..

7

7 / .. , .. ,
.. . 8- . .: , 2003.


I
C1
1. 1) )

5 1 10 1 9
1 5 2+5 7
1
;
=
=
+ =
= = 1 ; )
3 6
6
6
6
7 14
14
14

2
5 8 65 32 + 65 97
1
=
=8 ;
+5 = +
=
3
12 3 12
12
12
12
3 1 39 11 28
7 2 21 + 40 61
1
; )
=

=
=
=1 ;
+ =
11 13
143
143
20 3
60
60
60
2
1 47 8 329 120 209
104
;
3 1 =
=
=
=1
15 7 15 7
105
105
105
5 41 15 81
67
;

=
=
34 51
102
102
2
1
7
91 457 273 457
184
4
;
3 5
=

=
=
= 2
= 2
30
90 30 90
90
90
90
45
1
3 43 143 86 143
57
1
6 10 =

=
=
= 4 .
7
14 7 14
14
14
14

) 2
2) )
)
3) )
)
)

2. 1) ) 7 + 9 + 5,31 + 13,49 = 16 + 18,8 = 34,8;

) 62,7 + 0,07 + 8,31 + 5,79 = 62,77 + 14,1 = 76,87;
2) ) 8,31 (4,29 + 3,721) = 8,31 8,011 = 0,299;
) (8,21 + 9,73) 0,001 = 17,94 0,001 = 17,939.
3. 1) )
)
2) )
)
3) )

7 8 1 1 1
5 33
5 11 3
3
= ;

= = ; )
=
16 21 2 3 6
11 65
11 5 13
13
19 46
19 23 2
2
= .
=
23 57
23 19 3
3
1 9 13 48 48
1 1
16 3
16
3 3 =
=
= 12 ; ) 5 1 = =
= 8 ;
4 13 4 13
4
3 2
3 2
2
1 3
22 14
2 11 7 2
3 1 =
=
= 4 ;
7 11
7 11
7 11
5 10
5 21
573
3
1
=
= = 1 ;
: =
7 21
7 10
725
2
2

45
1
1
4 16 4 5
4
: 1 = : =
= ;
=
5 5 5 16 5 4 4 4
5 15
1
1 63 63 63 31 31
1
) 31 : 2 =
:
=

=
= 15 ;
2 31 2 31 2 63 2
2

5
5 9
4
4 9
4 1
4
( 9) = = 5 ; ) : 9 = : = =
;
9
9 1
5
5 1
5 9
45
4

14 14 5
) 14 : 2 = 14 : =

= 5;
5

5 1 14

4) )

5) ) 6,5 2,6 = 16,90 = 16,9; ) 5,3 7,7 = 40,81;

) 6,4 (1,3) = 6,4 1,3 = 8,32;

81 1000

= 90 ;
100 9
1515 100 303
) 0,1515 : 0,05 =

=
= 3,03 ;
10000 5
100
361 100 19
) 0,361 : 0,19 =

=
= 1,9 .
1000 19 10

6) ) 0,81 : 0,009 =

4. 1) ) 113 = 11 11 11 = 1331; ) 372 = 37 37 = 1369;

) (370) 2 = (37 10) 2 = 372 102 = 136900;
) (1100) 3 = (11 100) 3 = 113 103 = 1331000000;
2) ) (5) 3 = 53 = 125; ) (13) 2 = 132 = 169;
) (0,5) 3 = 0,53 = 0,125; ) (0,13) 2 = 0,132 = 0,0169;
3

4
9

1
5

3) ) =
2

4 4 16
33
27
3
; ) = 3 =
;
=
9 9 81
343
7
7

1
5

6
5

) 1 = 1 = =

1
5

6
5

) 1 = =

36
11
=1 ;
25
25

216
91
.
=1
125
125

5. , , ,
, .
1) 272; 7 7 = 19 ; 3) 1423; 2 2 2 = 8 ;
2) 532; 3 3 = 9 ; 4) 3113; 1 1 1 = 1 .
6. , : > 0. 1)

= 5 , = 5;

1
1
2) = : 10 = x
, x =
= 0,1 .
10
10

+ 77....7
22....2
....
....
7. 1) 77
12
37 + 22
12
32 = . : 99....9 .
123
100
100
100pa

....
, = 99
12
39
100pa

2) 55
....
....
12
35 + 88
12
38 = .
100

100

+ 55.....5
88.....8 ,
144...443

, = 144
....
12
343 ;
99

3) 55
.....
12
35 : 5 = .
100

, = 11
....
12
31 ;
100

.....
.....
....
4) 55
12
35 : 55
12
35 = . , = 100
12
301 .
100

50

49

C2

1
1 3 2 5 4
+ 5 :1 = + = 2 + 4 = 6 ;
2
4 1 3 1 5
2
1 1 32 16 5 32 5 27
) 10 5 : 3 =

=
=
=9;
3
3 5
3
3 16
3 3
3
1 1 32 16 3 16 3 16
6
3
2
) 10 5 : 3 =

=

=
=1 =1 ;
3 3 3
3 10 3 10 10
10
5
3
1 8
1
2 9 8 16 3
1 8 1 7
1
) 4 5 : 10 =

= 4 = = =3 ;
2 9
3
3 2 9 3 32
2 2 2 2
2

1. 1) ) 3 : 1

2) ) 0,7 1,3 + 5,1 : 0,17 = 0,91 + 30 = 30,91;

) 3,38 2,24 : 1,25 = 3,38 1,792 = 1,588;
) (3,38 2,24) : 1,25 = 1,14 : 1,25 = 0,912;
) 31,7 : 63,4 23,4 : 11,7 = 0,5 2 = 1,5.
2. 1) 3,12 + 2,92 = 9,61 + 8,41 = 18,02;
2) (5,3 (4,7)) 2 = (5,3 + 4,7) 2 = 100;
3) (1,37) + (1,35)) 3 = (0,02) 3 = 0,000008.
3. 1) (0,008 + 0,992) : (5 0,6 1,4) = 1 : (3 1,4) = 1 : 1,6 =

= 1:

16 1 10 10
=
=
= 0,625 ;
10 1 16 16

2) 13,5 9,1 (3,3) : (0,00013) = 13,5 3,3 9,1 : 0,00013 =

= 44,55 70000 = 3118500;

17
1
7
103 89 27 13 100
2 2,7 4 : 0,65 =


=
36
3
12
12 36 10 3 65
309 89 27 20 220 27 20 22 3 20 33 20
=

=
36
10 3
36 10 3
4
3
2
3
99 40 59
5
=
=
=9 ;
6
6
6

3) 8

8 5625 35 13 144 3
11 13
+ 1,44
=
+

=
15 10000 24 36 100 10
24 36
131 144 3 131 3 131 15 116 232
=

=
=
=
= 2,32 .
72 100 10 50 10
50
50 100

4) 1

4. 1) 1142; 4 4 = 16 6;
73; 7 7 7 = 49 7 3;
, 1142 73 6 3 = 3.
2) 1153 5;
152 5;
, 1153 152 5 5 = 0.
3) 173 3;
132 9;
, 173 132 13 9 = 4
(13, .. ).
3

25
= 50 ;
100
250
2) 3
= 0,75 ;
100

1. 1) 200

57 25 57
17

=
=1 ;
10 100 40
40
8 25
4)

= 0,02 .
100 100

3)

340 100
= 2000 ;
17
8,5 100
2) 17% 8,5. 100% , , x =
= 50 ;
17
0,051 100
3) 17% 0,051. 100% , x =
= 0,3 ;
17
2,89 100
4) 17% 2,89. 100% , x =
= 17 .
17

2. 1) 17% 340. 100% , , x =

3. 1) (8 : 16) 100% = 50%;

2) (8 : 800) 100% = 1%;

43
= 0,43 ;
100
75
) 75%
= 0,75 ;
100

4. 1) ) 43%

) 25% 0,25;
2) ) 0,5 50%;
) 0,37 37%;
) 0,7 70%;

3) (8 : 8000) 100% = 0,1%;

4) (8 : 0,8) 100% = 1000%.
) 60% 0,60 = 0,6;
) 11,4% 0,114;
) 1,35 135%;
) 1,2 120%.

5. :
m 1200 ;

m
100% = 20% ; m .
m
1)

m
20%
100% = 20% ; m =
m ;
100%
m

2) m
m = m m; m = 1200 240 = 960 .
3) 100% 20% = 80% (20% ) .
4)

m
240
100% =
100% = 25% .
m
960

6. 100%. 537000
1) 100% 537000; 102,5% ,
, x =

537000 102,5
= 537 1025 = 550425 .
100

2) 550425 537000 = 13425 .

7. 1) : 100% 150;
20%

x=

150 20
= 30 .
100

2) : 150 30 = 120 () .
100% 120; 25% ,

x=

120 25
= 30 () .
100

3) 2 : 30 + 30 = 60 .
4)

60
100% = 40% .
150

20
1
100% =
100% = 5% ;
2
20
20
0,2
100%
2)
100% =
= 2500% .
0,04
0,2 3

8. 1)

9.
: + 0,2.
: + 0,2 0,2 ( + 0,2) = 0,04 =

= x

4
x.
100

: 4%.
6

4
1. 1) 6,8 + 3,2 = 3,6; 3,2 + 3,2 = 0;

1
4 16 20 + 48 68
8
=
1 + 3,2 = +
=
=4 ;
3
3 5
15
15
15

2) 5 (2,6) = 13; 5 0 = 0; 5 1 = 5;

52

8
5 38
38
2
=
=
= 12 ;
15
1 15
5
3

3) 12 (1) 7 = 12 7 = 19; 12 0 7 = 0 7 = 7;
12 (7,6) 7 = 91,2 7 = 98,2;
12 0,05 7 = 0,6 7 = 6,4;
4) 3 1,5 4 = 3; 3 1,5 (2) = 6;

3 1
1
3 1,5 = 3 + = 3,5 ;
2 3
3
3 1,5 0,8 = 3 1,2 = 1,8.
2.

5 3
3 5
(3 5)

3
18
18
54

2
13
13
26

1
8
8
8

0
3
3
0

1
2
2
2

2
7
7
14

3
12
12
36

3. 1) ) 8 (7) 11 (3) = 56 + 33 = 23;

3
8 11 0,6 = 6 6,6 = 12,6 ;
4
) 5 0 4 12 = 0 48 = 48;
5 (1,2) 4 3,25 = 6 13 = 19;
2) ) 8 (4) + 3 10 + 1 = 32 + 30 + 1 = 1;
8 (6,5) + 3 4

2
+ 1 = 52 + 14 + 1 = 37 ;
3

) 1 5 12 3 (16) = 1 60 + 48 = 11;
1 5 (11) 3 (11) = 1 + 55 + 33 = 89;
3) ) (1,7 1,3) (1,7 + 1,3) = 0,4 3 = 1,2;
) 2 0,3 (0,6 3 0,2) = 2 0,3 0 = 2;
)

2,8 + 2 0 2 2,8 5 0 2,8 5,6 5,6 5,6 0

=
= =0.
3
6
3
6
6
6

4. 2, , , .. , n 2 (2).
= 20; n = 500.
S , S = na2 (2);
S = 500 202 = 500 400 = 200000 (2).
7

5. 1) a + b;
2) a V1;

4) a V1 + b V2;
5) V =

aV1 + bV2

; V =
.

a+b

3) b V2;

1
;
7
1 3
3 0 = 0 ; 3 = ;
7 7

6. 1) 3; = 0 x =

2) 2,7; = 10 = 1;
2,7 10 = 27 ; 2,7 1 = 2,7 ;

1
= 2;
3
3 1
3 10
3 + 5 = + 5 = 1 + 5 = 4 ;
10
3
10 3

3) 0,3 + 5; x = 3

0,3 2 + 5 = 0,6 + 5 = 5,6 ;

4) 3 + 0,1; = 0,3 = 0;
3 0,3 + 0,1 = 0,9 + 0,1 = 1 ;
3 0 + 0,1 = 0 + 0,1 = 0,1 .
7. 1)
2)
3)
4)

x + y z = (x + y) z = 5 (8) = 5 + 8 = 13;
2z (x + y) = 2 (8) 5 = 16 5 = 21;
x 5z + y = (x + y) 5z = 5 5 (8) = 45;
3 (x + y) + 2z = 3 5 + 2 (8) = 15 16 = 1;

8
z
2
8
=
=
=2 ;
x + y + z 58 3
3

5)

6) z (x + y + 5z) = 8 (5 + 5 (8)) = 8 (5 40) = 8 (35) = 280.

8. ) a3b = a 100 + 30 + b ; ) 5 xy = 500 + 10 x + y ;
) pp3 = 100 p + 10 p + 3 = 110 p + 3 .
C5

1 1 8 + 3 11 1 1 9 + 2 11
; + =
;
+ =
=
=
3 8
24
24 2 9
18
18
11 11
1 1 1 1
>
, .. 18 < 24. , + > + .
18 24
2 9 3 8
3 5
5 3 2
) < 0 ; = > 0
11 7
7 7 7
5 3
3 5
, > ;
7 7
11 7

1. 1) )

1 1 1 4+3 7 7 1 76 1
=
> 0;
=
; + =
;
=
12
12
2 3 4
12
12 12 2
1 1
, + > 0,5 ;
3 4
2 1
5 1 10 + 1
9
3
) 1 + = + =
= = = 1,5 ;
3 6
3 6
6
6
2
2 1
1,5 (1,6) = 1,5 + 1,6 = 0,1 > 0, , 1 + > 1,6 .
3 6

2) ) 0,5 =

2
3

2. 1) 2 0 + 5 = 5 2 1 + 5 = 5

10
,
3

, = 0 ;
2) 3 3 1 = 0 3 3 (1) = 6,
, = 1 ;
3) 3 (0,3) + 5 0,6 = 2,1 3 1,2 + 5 (0,3) = 2,1,
, .
3. 1) ) t < 5.
, t = 1; 0; 4 ;
t = 5; 5,1; 11 ;
) 11,3. = 11,3; 0; 11,3 ;
= 20; 18; 11,4 ;
) m 0. : m = 1; 2; 0. : m = 0,0001; 10; 100;
2) ) 5 > x 4. : = 4; 4,2; 4,99. : = 5,1; 3; 0;
) 0,01 < a < 0,02;
: = 0,011; 0,015; 0,0199. : = 1; 0,02; 12;
) 0,7 < 0;
: = 0,15; 0,6; 0,59. : = 0; 0,72; 0,1.
4. m1 = m1 + m1; m2 = m2 + m2;
m1, m2 ; m1, m2 ;
m1, m2 ; w1, w2 . ;

3
100% = 60% ;
5
13
m2 = 13 + 7 = 20 ; w2 =
100% = 65% ;
20

m1 = 3 + 2 = 5 ; w1 =

, w2 > w1.
5. 1) 2,8 0,16 > 2,8, .. 2,8 (1 0,16) > 0;
2) 0,16 < 2,8 0,16, .. 0,16 (2,8 1) > 0;
3) 2,8 0,16 > 2,8, .. 2,8 (0,16 + 1) > 0;
4) 0,37 :

1
1
> 0,37 : 5 , .. 0,37 5 > 0,37 ;
5
5
9

5) 0,37 > 0,37 :

1
, .. 0,37 > 0,37 3, .. 0,37 (3 1) > 0;
3

6) 86 : (3,4) < 76 : (3,4) , .. 86 : 3,4 > 76 : 3,4;

86 : 3,4 < 76 : 3,4.
6. 1)

1
1
8
7
5
;
; ; ; ;0;
13 13 13 13 100

7. 1) 1,09; 1,009; 0; 1,23; 1,24;

2) (0,1) 3; (0,1) 2; 0,1.

2) (0,2) 2; (0,2) 3; 0,2.

8. ;
1- : + 0,25 ;
2- : + 0,25 + 0,2 ( + 0,25) = + 0,25 + 0,2 + 0,05 =
= + 0,25 + 0,25;
: , 0,25 + ;
: + 0,25, + 0,25 + 0,25,
, .
6
1. 1) ) (6,83 + 3,17) + (7,81 + 8,19) = 10 + 16 = 26;
1
3
7
1
) (7 + 15 ) + (13 + 17 ) = 23 + 31 = 54 ;
4
4
8
8
2) ) (925 825) + 527 = 100 + 527 = 627;
) (5,37 + 4,37) + 9,29 = 8,29;

28 5
2 19
) 13,5 = 27 ;
) ( ) 3,9 = 39 ;
1 14
19 1
3 11 21 17
4) ) ( ) ( ) = 1 1 = 1 ;
11 3 17 21
1
1
7 31 2 13
)

= .
= 1
31
7
13
20
10
10

3) ) (

1
5
= 35 + 1 = 36 ; 2) ) 12 3 + 12
= 36 + 5 = 41 ;
5
12
1
1
) 13 10 + 13
= 131 ;
) 8 9 + 8 = 72 + 2 = 74 .
13
4

2. 1) ) 5 7 + 5

1 2 3 4 5 6 7 1
= ;
2 3 4 5 6 7 8 8
6 7 13 12 11 10 9 8 13
;
) =
7 6 12 11 10 9 8 7 7
13 12 11 10 9 8 7 6 13
=
.
12 11 10 9 8 7 6 7 7

3. )

10

1 1 1 1 1 1 1 1
1
= ; =
; =
;
2 3 6 3 4 12 4 5 20
1 1
1 1 1
1 1 1
1
=
; =
; =
;
5 6 30 6 7 42 7 8 56
1 1
1
1
1
1 1 1 1 1 1 1 1
+ +
+
+
+
= + + +
6 12 20 30 42 56 2 3 3 4 4 5 5
1 1 1 1 1
1 1 4 1 3
+ + = S ; : S = =
= .
6 6 7 7 8
2 8
8
8

4.

5. 1) ) 5 822 = 5 2 411 = 4110;

) 5 412 = 5 2 206 = 10 206 = 2060;
) 5 (724) = 5 2 362 = 3620;
2) ) 822,2 5 = 411,1 10 = 4111; ) 43,6 5 = 21,8 10 = 218;
) (0,626) 5 = 0,313 10 = 3,13.
7
1. 1) ) 2,8 5 = 14; ) 3,5 4 = 3,5 4 = 14;
) 3,6 0,8 = 2,88;
) 8 (12) = 8 (12) = 96;
2) ) 8 (3) = 8 (3) = 24;
) 3,5 2 = 3,5 2 = 7;
) 0,25 8 b = 0,25 8 y b = 2by;
3
7
3 7
1
) p q = p q = pq .
7
9
7 9
3
2. 1) ) 2 + 3 = 5;

) 7 15 = 8;
) 17b 3b = 20b;
) 2,1 + 7 = 4,9;

) 0,8 = 1,8;
1
5
) 2 = ;
3
3
1
1
) + = 0,7 ;
2
5
5
1
) b b = b .
6
6

) 2,5 + = 1,5;
2) ) 8b + 12b 21b + b = (8 + 12 21 + 1) b = 0 b = 0;
) 13 + 12 + 40 18 = (13 + 12 + 40 18) = 21;
) 3 = (1 1 1 3 1 1) = 8;
) 4,14 + 8,73 + 5,8 = (4,14 + 8,73 + 5,8 1) = 17,67.
3) ) 10 b + 7b = (10 1) a + (7 1) b = 9a + 6b;
) 15 15 + 8 + 4 = (4 15) + (8 15) = 11 7;
) 0,3 + 1,6 0,3 0,4 = (0,3 0,3) + (1,6 0,4) =
= 0 + 1,2 = 1,2;
) + + 4 = (1 1) + (1 1) + 4 = 0 + 0 + 4 = 4;
11

) 5 + 4 b 6a = 5 + (1 + 4 6) a b = 5 3a b;
) 1,2 + 1 0,6 0,8 0,2 = (1,2 0,2) 0,6 + (1 0,8) =
= 0,6 + 0,2.
3. 1) ) + (a + b) = c + a + b;
) c (a + b) = c a b;
) (a b) = c a b;
) c (a + b) = c + a b.
2) ) (a b) (c d) = a b c + d; ) (a b) + (c d) = a b + c d;
) x (a b) + (c d) = x a + b + c d;
) 10 (a b) (c + d) = 10 a + b c d.
4. 1) ) 3b + (5a 7b) = 3b + 5a 7b = 5a 4b;
) 3q (8p 3q) = 3q 8p + 3q = 8p;
) 5 (11 7) = 5 + 11 7 = 11 2;
) (8 4) + 4 = 8 + 4 + 4 = 8 8;
2) ) (2 + 3) + (7 2) = 2 + 3 7 2 = 10;
) (11a + b) (12a 3b) = 11a b 12a + 3b = 2b 23a;
) (5 3b) + (3b 11) = 5 3b + 3b 11 = 6;
) (5a 3b) (2 + 5a 3b) = 5a 3b 2 5a + 3b = 2;
3) ) + ( 10) (12 + ) = + 10 12 = 22;
) (6 8) 5 (4 9) = 6 + 8 5 4 + 9 = 10 12;
) (1 9) (22 4) 5 = 1 9 22 + 4 5 = 31;
) 5b (6b + a) (a 6b) = 5b 6b a a + 6b = 5b 2a.
5. 1) ) 3 (8 4) + 6 = 24 12 + 6 = 30 12;
) 11 + 5 (8 ) = 11 + 40 5 = 6 + 40;
) 2 ( 1) 2 + 12 = 2 2 2 + 12 = 10;
) 16 + 3 (2 3) + 8 = 16 + 6 9 + 8 = 22 ;
2) ) 7 2 (3 1) = 7 6 + 1 = + 2;
) 4 (3 + 2) + 8 = 12 8 + 8 = 12;
) 3 17 11 (2 3) = 3 17 22 + 33 = 36 39;
) 15 5 (1 ) 6 = 15 5 + 5 6 = 10 .
6. ) ( (2 4)) = + (2 4) = 2 4;
) 7 (( ) + 3) = 7 ( ) 3 = 7 + 3 = 8 4;
) 4 (3 (2 ( + 1)) ) = 4 3 + (2 ( + 1)) =
= + 2 ( + 1) = 3 1 = 2 1;
) 5 (2 ((b c) 2b)) = 5c 2c + ((b + c) 2b) =
= 3c + (b c) 2b = 3c + b c 2b = 2c b.
7. ) 0,6 + 0,4 ( 55) = 22 = 8,3 22 = 30,3;
) 1,3 (2 ) 16,4 = 1,3 12 16,4 = 15,6 16,4 = 0,8;
) 1,2 ( 7) 1,8 (3 ) = 1,2 8,4 5,4 + 1,8 = 3 13,8 =
13
= 3 13,8 = 0,8 ;
3
1
2
7
7
23
23
) 2 (a + 6) 7 (3 a ) = a + 6 3 + a = 10a 9 =
3
3
3
3
3
3
= 7 9 = 16.

12

8
1. ) 8 = 24;

) 50 = 5;
) 18 = 1;
2
;
8
3
) x = 1 ;
5
1
) 5 x = ;
5
1
) x = 6 ;
6
3
2
;
) x =
7
14
) 0,81 = 72,9;

) 3x =

2. ) 3 = 0;
) 3 = 6;
) 3 = 12;
3
) 3 x = ;
17
10
;
) 3 x =
3
2
) 3 x = 2 = 2,4 ;
5

24
=3;
8
5
1
x=
= ;
50
10
1
x=
;
18
2
1
x = : ( 3) = ;
8
12
3
x =1 ;
5
1
x=
;
25
x=

= 36;
2 7
1
= ;
14 3
3
= 72,9 : 0,81 = 90.
x=

= 0;
= 2;
= 4;
1
x=
;
17
10
;
x=
9
= 0,8.

3. ) 3 = 3 (11) = 33; ) 5 = 5 0 = 0; )
4. ) S = V t;
S
V= ;
t

S
;
V
) J R = U;
U
U
J = ; R= ;
R
J
t=

2
2
= 14 = 4 .
7
7

) mg = P;
P
m= ;
g

g=

P
.
m

13

1
= 8 = 56
7
0 = 8, 0 = 0, 0 = 8 ,
, , = 0 .
8
2) = 8 = , , , 0.

: = 0. 0 = 8 0 = 8 .
8
8
3) = 8 = . x < 0, , < 0 , , < 0.

5. 1) (4) = 8 = 2; a

9
1. 1) ) 3 + 7 = 0;
3 = 7;
7
x= ;
3
) 13 100 = 0;
100 = 13;
2) ) 7 4 = 16;
6 = 12;
= 2;
) 13 5 = 8 2;
5 = 3;
5
= ;
3
) 4 + 15 = 6 + 17

2 = 2;

= 1
3) ) 5 + (3 7) = 9;
5 + 3 7 = 9;

) 0,5 + 0,15 = 0;
0,5 = 0,15;

= 0,3;
) 8 0,8 = 0;
= 0,13; 0,8 = 8; = 10;
) 1,3 11 = 0,8 + 5;
0,5 = 16;
= 32;
) 0,71 13 = 10 0,29;
= 23;

) 8 + 0,73 = 4,61 8;
97
;
16 = 3,88 =
25

) 48 = 11 (9 + 2);
48 = 11 9 2;
13
8 = 16; = 2;
9 = 39; = ;
3
) 13 (5 + 11) = 6;
) 3 (5 ) 11;
13 5 11 = 6;
3 5 + = 11;
2 = 11;
4 = 16;
2
= 4;
= .
11
4) ) (7 + 1) (6 + 3) = 5; 7 + 1 6 3 = 5; = 7;
) (8 + 11) 13 = 9 5; 8 + 11 13 = 9 5; 3 = ; = 3;
) 2 = (3 5) (7 4); 2 = 3 5 7 + 4;14 = 7; = 2;
) 8 + 5=119 + (7 3); 8 + 5= 119 + 7 3; 11 = 121; = 11.
14

2. 1) 5t + 11 = 7t + 31; 20 = 2t; t = 10;

2) 8t + 3 = 3 (5t 6); 8t + 3 = 15t 18; 21 = 7t; t = 3;
3) 2 (5t + 1) = 10t + 18; 10t + 2 = 10t + 18;
0 = 16 , , t;
4) 0,25t 31 = 0,25t 18 + 5; 0 = 18 , t;
5) 13t 7 + 8 = 12t + 1; t = 0;
6) (1,5t 37) (1,5t 73) = 36; 1,5t 37 1,5t + 73 = 36;
36 = 36 , , t.
3. ) (5 3) + (7 4) = 8 (15 11);
5 3 + 7 4 = 8 15 + 11; = 0;
) (4 + 3) (10 + 11) = 7 + (13 4);
4 + 3 10 11 = 7 + 13 4; 28 = 2; = 14;
) (7 5) (8 4) + (5 + 6) = 8;
7 5 8 + 4 + 5 + 6 = 8;
3
4 = 3; x = ;
4
) 3 2 + 4 3 + 5 5 = 12 + 7;
0 = 17; = 0.
4. 19 (2 3) = 19 (5 + 6)

2 x 3 5x + 6
=
;
11
11

: 2 3 = 5 + 6;
9 = 3;
= 3.
5. 3 + 7 = 3 + 11 5 = 6 || + 1 = 0.
10
1. ,
63 ;
+ 63 = 657;
2 = 720;
= 360 () ;
360 63 = 297 () .
2. , 111 ;
2 = 111 ;
3 = 111;
= 37 () ;
111 37 = 74 () .
3. , ,
4 , ;
+ 4 = 40; 5 = 40;
= 8 ();
4 8 = 32 () .

15

4. 3- ; 3 1-
+ 5000 = 3; 2 = 5000;
= 2500 (.) 3- ;
3 2500 = 7500 (.) 1- .
5. ;
+ 12 ;
3 ( + 12) = 5; 36 = 2;
= 18 / ;
18 + 12 = 30 (/) .
6. ; 84 ;
( 1) 3 = 84 + 1;
3 3 = 85 ; 4 = 88;
= 22 ;
84 22 = 62 () .
:
(84 1) 3 = + 1; 249 3 = + 1;
4 = 248;
= 62 () ;
84 62 = 22 () .
7. ;
= 22 + 0,5; 0,5 = 22;
= 44 () .
8. ;
+ 30 .
. 4
(1 , 3 ) . , :
4 = 3 ( + 30). = 90 (/) .
11
1.

4
P

I
-4

1 O
C
-2 -1 0
1 2
-1
O1
E

-2
-3

V
H
3

D
L1

-4
-6

16

-6

R1

2. (3; 2); (2; 4); (3; 0); D (0; 1); E (3; 4); F (2; 2); H (4; 3);
K (4; 0); L (6; 1); M (0; 5); O (0; 0).
3. 1) (1; 1); (2; 3); (1; 1); D (2; 3); E (1; 1); F (4; 4);
G (3; 1); H (1; 1).
2) O (0; 0); M (1; 0) ; (0; 0); N (0; 1) .

D
C
-4

-2 -1

E
F

N A

0
-1

M
1 2

-4

4. 1) (4;1); (4;1); (1;3,5); D (1;3,5); (4;1); F (4;1);

G (1; 3,5); H (1;3,5);
2) : (4; 0) M (4; 0). : N (0; 3,5) N (0; 3,5)
5. 1) : (1,25; 0);
5
: N 0; ;
7

2) M (0; 1,5) M (0; 1,5).

C
1
-4

0
-1

1
1

3
D

E
0

B
A

1
M

6. 1) ; ; ;
2) K ; L .

17

 12
1. 1) = 4 (3) 8 = 20; = 4 0 8 = 8;
= 4 1 8 = 4; = 4 6 8 = 16;
1
2
6
2) y = 1 = 1 ; y = 1 = ;
3
3
3
0
1,5
y = 1 = 1 ; y =
1 = 1,5 ;
3
3
2
2
3) = (3) = 9; = 0 = 0; = 32 = 9; = 4,52 = 20,25.

1
2.
0,8 0,4
1,2
= 0 = 0,8;
= 0 = 2.

0
0,8

3. 1) 12 = 2,5;

x=

12
= 4,8 ;
2,5

1
0,4

2)

2
= 4x + 3 ;
3
7
4x = ;
3

x=
4. 1) b = |5| 4 = 5 4 = 1;
b = |0| 4 = 4;
b = |4| 4 = 4 4 = 0;
2) b = |5 + (5) | = |0| = 0;
b = |5 + 0| = 5;
b = |5 + 4| = 9;

2
0

7
.
12

3) |a| + 5b = 4b + 1;
b = 1 |a|;
b = 1 |5| = 4;
b = 1 |0| = 1;
b = 1 |4| = 3.

5. 1) = ;
1 1
(0; 0); (1; 1); ; ;
3 3
2) = 2;
1
(0; 0); (1; 2); ;1 ;
2
3) = 2 3;
3
(0; 3); (1; 1); ;0 .
2

6. 1) = 3 1 3 = 0 =1;
2) = 12 = 1, = 1;
= 3 0 3 = 3 = 0;
= 7, = 0;
1
= 7, = -6.
y = ( 6 ) 3 = 5 = -6;
3

18

3
0,4

 13
1. 1)

0
4

2
2

2)

0
6

3
12

12

2
1
0

= - + 4

= 2 + 6

= + 4

= 2 + 6

2. 1) )

=+1

= -3 - 3
6

1
0

-3

-3

= 4 - 6

2
1
0

1 2

-6

19

2) )

3
1
-4

y=

= 0,4 + 2

4
2
1
0

y=

2
3

x 1

1
0
-3

20

1
2

x +1

3) ) , ) , )

=5

1
0

= -4
-4
-6,5

= -6,5

3. = 4 6;
1) = 1: = 4 1 6 = 2;
= 1: = 4 (1) 6 = 10;
= 0: = 4 0 6 = 6;
= 2: = 4 2 6 = 2;
2) 4 6 = 3;
9
4 = 9; x = ;
4
4 6 = 1; 4 = 5;
5
x = ; 4 6 = 0;
4
4 = 6; = 1,5;
4 6 = 2; 4 4; = 1.
4. 1)

2)

= 2-

= - + 4
= 0,5 + 1

=-2

1
0

(2; 2)

(2; 0)

21

3)

=-1

y=

1
x 1
3

1
0
-1

(0; -1)

5. 1) )

y=
1

1
-3

-1

2) )

y = 1

x
3

-2

= 2( 3), 0

1
0

2x 6
3

y=

1
(8 x ), x 0
2

5
4

1
0

-6

22

6. 1)

2)

=+4

6
=4

-1 0

1
0

=+1

=1

3, x < 0
7. ) y =
;
1, x 0

2, x > 0
) y =
2, x 0

=3
=2

2
1

1
0

= -1

-2

= -2

14
1. 1) = 2,5;

2) = 2

0
0

2
5

0
0

2
4

= 2,5

= -2

1
0

1
1

-2

23

2. 1) )

= 4

= 1,5
3
1

1
-2

y=

1
3

1
0

2) )
= -3

y=

1
-1 0

4
x
3

-4

24

= -0,4

1
0

-2

3. = 3
1) = 1: = 3 1 = 3;
= 2: = 6;
= 1,5: = 4,5;
= 1: = 3;
2) = 3 = 0;
= 0;
= 3 = 2;
2
x= ;
3
= 3 = 2;
2
x= ;
3
= 3 = 3; = 1.
4. = 3 y =

1
x.
3

1
y= x
3

= 3

3
1

-3

25

5. (1) = 2;
(2) = + 2;
(3) = 2.
9
9
; y= x;
2
2
7
7
) y = kx; 7 = 3k; k = ; y = x .
3
3

6. ) y = kx; 9 = 2k; k =

7. 1) ) , ) , )

y=

y=

x
0,5

2
1
y=

2) )

2
5

0
-1

= -

2
1

1
-2

-1 0

-2
-4
= 2

= -2

15
1.

1)
26

1
3

0
1

2
3

3
7

0,5
2

2,5
6

2,5
4

2
3

2)

4
0,5

1
1

1
2

5
4

5
1

1
2

4
3,5

3
0

3)

4
30

1
15

0
10

3
5

2
20

1
5

2
0

5
15

2. ( 5);
1) : (3; 0). : N (0; 1,5);
2) = 2; 0; 1;
3) = 4; 7; 11.
3. 1) ) 50 ; ) 5 ;
2) ) 45 ; 35 ; 25 ;
) 8 ; 14 ; 20 ;
3) ) 6 ;
) 5 ;
4) ) ; ) .
4. 1) 3 ;
2) 5 ;
3) ) V = 50 5x;
) V = 5 + 3x;
x ; V .
16
1. 1) k1 = k 2 = k 3 =

y=

1
3

1
;
3

x +1

1
0
-1

-2
-3

y=
y=

1
3
1
3

x
x2

2) ;
3) M1 (3; 0); N1 (0; 1); Mi ;
M2 (6; 0); N2 (0; 2); Ni ;
M3 (0; 0); N3 (0; 0).
27

2.

1
-1 0
= -2

1) M1 (2; 0); N1 (0; 2); Mi

;
M2 (1; 0) N2 (0; 2) Ni
M3 ; N3 (0; 2)
2) 3 (0; 2).

-2

=2
= -2 2

3. M1 (2; 0); N1 (0; 6);

M2 (2; 0); N2 (0; 6);
= 3 6
M3 (2; 0); N3 (0; 6);
M4 (2; 0); N4 (0; 6).

= -3 6

) = 3 6
= 3 + 6;
1
-2

= 3 + 6

) = 3 6
= 3 + 6.

= -3 + 6
-6

4. ) 2 4 = 4 + 2;
= 2 4

2
1
0
-2
-4

28

2
L

= -4 + 2

6 = 6;
= 1;
= 2 1 4 = 2;
L (1; 2) ;
) 2 3 = 2 + 3;
0 = 6 , ,
.

2
x+b ;
3
2
3 = ( 6 ) + b ;
3
b = 1;
2
y = x +1 ;
3

5. ) y =

) = 4 + b;
7 = 4 2 + b;
b = 15;
= 4 + 15.

6.

k =

1
;
2

1
x+b ;
2
4 = 3 + b;
b = 1;
1
y = x + 1 ;
2
1
k1 = k 2 = ;
2
1
y = x+b ;
2
5 = b;
1
y = x+5.
2
y=

y=

1
2

x +1

1
0

7. ) . 8
1
y= x.
4
.
) . 9
= 3.
k < 0,
0
90, .. .
) . 10
, . 9, .
17
1. 1) ) l = 30 + 4 5 = 50 ();
) l = 30 + 4 8 = 62 ();
2) , .
, k = 4, b = 30;

) l = 30 + 4 3 = 42 ();
) l = 30 + 4 0 = 30 ();

29

3)
l
l = 30 + 4m

62

60
50
40
30
20
10

m
0

4) ) l = 41,6 ();
) l = 52 ();
) l = 37,2 ();
) l = 30 ();
2. 1) ) 38 ();
) 37 ();
2) , .
1
k = ; b = 40 ;
50
3)

5) ) m = 7,5 ();
) m = 3,75 ();
) m = 7 ();
) m = 0 ();
) 36 ();
) 40 ();

h
40
30

h = 40

28

m
50

20
10
0 100 200 300 400 500 600

4) ) h = 39,5 ();
) h = 38,2 ();
) h = 36,8 ();
) h = 35,8 ();
) h = 40;
6. ) 0,5 ; 0,5 ;

30

5) ) 350 ;
) 100 ;
) 250 ;
) m = 0 ;
) 1 .

 18
3

1
1
) =
;
4
64

1. 1) ) 34 = 81;

) (0,6) 2 = 0,36;

2) ) (8) 2 = 64;

243
1
) 1 =
;
32
2
) (1) 7 = 1;
4

) (0,5) 3 = 0,125;
3) ) 72 = 49;

1
1
) =
;
3
81

) (0,1) 4 = 0,0001.

8
2
) =
;
27
3
2. ) (9,2) 2 = 9,22 > 0;
) (13,6) 3 = 13,63 < 0;

) 475 < 0;
) 7,22 < 0.

3. 21; 23; 25; 27;

0,11; 0,13; 0,15;
2

1 1
; ;
2 2
(3) 4; (3) 3; (3) 1.
4. 1) ) 0,1 3600 = 360;

0,4 3 0,064 2
=
= 0,00128 ;
50
100

3
2

2) ) =

1
9
=2 ;
4
4

) 3

1
1
= ;
27
9

) 0,2 16 = 3,2.
) 7 + 72 = 7 (1 + 7) = 56;

3
) 6 = 729 ;
) 0,25 = 0,00032;
2

3) ) 216 64 = 152; ) 10000 125 = 10125; ) 1 1 = 2.

5. 1) ) 20796,872; ) 530,8416; ) 1,25 = 2,48832;
2) ) 13,08; ) 6,5536; ) 10,209 + 9,61 = 19,819.
6. 1)
2)
3)
4)

0,32 + (0,7) 2 = 0,09 + 0,49 = 0,58;

(6,4 5,9) 2 = 0,25;
1,52 0,62 = 2,25 0,36 = 1,89;
(1,7 + 0,3) 2 = 1,42 = 1,96.
31

7. 1) )

2) )

8
9 4 1 1
= ; ) 3 10 3 3 64 = 40 ;
16 3 4 2
10

8
1
16 16
+ 27 = 37 ; ) 3 4 3 1 : 3 = 24 1000 = 1024 .
:
10 100
10
3

8. 1) ) 4,1 5,66 < 0; ) 3,33 : 5,7 > 0;

2) ) 4,82 1,24 < 0; ) 2,74 (6,45) > 0.
9. 1) ) 6,54 > 2,43, (6,5) 4 > (2,4) 3;
4,7 5
;
) 0 > 4,9 0,82 4,75 : (0,63) =
0,6 3
, (4,7) 5 : (0,6) 3 > (0,8) 2 (4,9);
2) ) (0,2) 6 > (0,2) 10, .. 0,26 < 0,210;
) (1,5) 7 < (1,5) 9, .. 1,57 > 1,59.
19

16 16
7
;
=1 ;
25 9
5
1
;
2) 125; 0,001; 27;
64
3) 810; 2,5; 14,4;

1. 1) 49; 121; 0,64;

4) 108; 0,032; 62,5;

5) 116; 28; 72;
6) 3 2 = 2 ( 1);
0,063; 36 (7) = 252.

2.

1)

2
2
2
4

2)

3
0,13
3 + 10

5
25
25
21
4
64
6,4
54

2,5
6,25
6,25
2,25

0
0
0
4

0,3
0,027
0,0027
9,973

0,3
0,09
0,09
3,91
1
1
0,1
9

1
1
1
3
0
0
0
10

3. 1) (12 (0,5)) 2 = 36; ((14) (1)) 2 = 196;

3

6
2) = 64 ; 0;
1,5
3) (0,7 + 0,3) 4 = 1; (11 + 6) 4 = 625;
4) (10 + 14) 3 = 64; (1,1 0,9) 3 = 0,008.
4. 1) 32 = 9 < (3) 2 = 9; (5) 2 = 25 < 52 = 25;
02 = 0 = (0) 2 = 0;
2) (a) 3 = a3 , = 10; 2; 0.

32

12
144
144
140
9
729
72,9
739

5. 1) ) 2 0;
) 2 0;
) 2 + 4 > 0;
) 2 2 < 0;
) ( + 5) 2 0;

2) ) 2 + 2 0;
) 2 + 2 + 10 > 0;
) ( ) 2 0;
) 5 ( + ) 2 0.

20

) 3 3 = 6;
1. 1) ) 7 4 = 11;
2
3
) = ;
) 38 34 = 312;
2
3
6
2) ) b b b = b ;
) 6 3 7 = 16;
) (7) 3 (7) 6 (7) 9 = (7) 18 = 718.
2. 1) ) 8 : 4 = 4;
) 10 : 9 = 1 = ;
) 6 : = 5;
) 5 : 5 = 0 = 1;

2) ) 214 : 28 = 26;
) (0,1) 20 : (0,1) 6 = (0,1) 14;
) (0,5) 16 : (0,5) 8 = (0,5) 8 = 0,58.

3. 1) 3 7 = 10;
2) = 2;

3) 12 : 6 = 6;
4) 11 : 5 = 6.

4. 1) 2 8 : = 9;
2) 5 : 2 : 2 = ;

3) 15 : 5 = 11;
4) 10 : 6 4 = 8.

5. 1)
2)
3)
4)

1015 107 : 1019 = 103 = 1000;

78 : 7 : 75 = 49;
(3) 5 (3) 3 : (3) 7 = 3;
(0,2) 8 (0,2) 2 : (0,2) 4 : (0,2) 3 = 0,23 = 0,008.

6. 1) (11) 9 (11) 8 = (11) 9+8 = (11) 17 = 1117 < 0;

2) (6) 4 (6) 10 = (6) 4+10 = (6) 14 = 614 > 0;
3) (14) 25 : (14) 8 = (14) 258 = (14) 17 = 1417 < 0.
7. 1) am an = am+n;
2) bn b2n = b2n+n = b3n;
3) yn : y3 = yn3;

4) 10 : ym = y10m;
5) c cn = c1 cn = cn+1;
6) cn : c1 = cn1.

8. 1) ) 10 : (10 : 5) = 10 : (5) = 5;
) 18 (9 : 7) = 18 2 = 20;
) 6 : ( 5) = 6 : 6 = 66 = 0 = 1;
2) ) (4 3) : (3 2) = (7) : (5) = 2;
) (16 8) : 4 2 = 8 : 4 2 = 4 2 = 6.
9. 1) (83) (811) = 814 < 0;
2) (6) 12 = 612; 612 64 = 616 > 0.
10. 1) xn+6 = xn x6; xn+6 = xn+8 : x2;
2) a3n = a2n an; a3n = a4n : an;
3) yn = yn1 y1; yn = y3n+2 : y2n+2.

33

 21
1. 1) ) (ab) 9 = a9 b9;
) (xyz) 7 = x7y7z7;
) (0,1) 4 = 0,00014;

2) ) (2) 3 = 83;
) (0,4) 2 = 0,162;
) (3) 5 = 24355;
4

) (2) 4 = 1644;

16 4 4 4 2

a b c = abc .
81
3

1 3 3 3
1

) xyz =
x y z ;
3
27

2. 1) ) (1 ) 2 = (1) 22 = 2;
) (1 ) 8 = (1) 88 = 8;
2) ) (1 ) 3 = (1) 33 = 3;
) (1 ) 9 = (1) 99 = 9;

) (1 ) 100 = (1) 100100 = 100;

) (1 ) 2n = (1) 2nx2n = x2n;
) (1 ) 71 = (1) 7171 = 71;
) (1 ) 2n+1 = (1) 2n+1x2n+1 = x2n+1.

3. 1) ) 55 = () 5;
) 36a2b2 = (6ab) 2;
) 0,00133 = (0,1) 3;
2) ) 3 = () 3;
) 83 = (2) 3;
) 325b5 = (2ab) 5;
3) ) 55z5 = (xyz) 5;
) 0,027a3b3c3 = (0,3abc) 3;
3

)
4. 1)

1 3 3 3 1

x a z = xaz .
64
4

23 = (5 2) 3 = 1000;
4

1
1

2) 20 4 = 20 = 625 ;
4
4

3) 0,53 603 = (60 0,5) 3 = 303 = 33 103 = 27000;

4

6 5
2
4) 1,2 4 1 = = 16 .
3
5 3
5. 1) ) (5) 2 = 10;
) (4) 3 = 12;
2) ) (2) 3 = 23 = 6;
) (3) 2 = 6;
6. 1) (5) 5 = 25;
2) (5) 2 = 10;

34

) (10) 10 = 100;
) (m) 2 = x2m;
) (4) 2n = a8n.
3) (an) 3 = a3n;
4) (a2) n = a2n.

7. 1) ((2) 2) 2 = (4) 2 = 8;
2) ((3) 3) 3 = (9) 3 = 27;
3) ((2) 3) 4 = (6) 4 = 24;

4) (() 3) 2 = (3) 2 = (3) 2 = 6;

5) ( () 2) 3 = (2) 3 = (2) 3 = 6.

8. 1) ) 85 = (23) 5 = 215;
) (162) 3 = ((24) 2) 3 = 224;
2
2 2
4
2) ) 4 = ((2) ) = (2) ;
) ((4) 3) 2 = (43) 2 = (43) 2 = (( (2) 2) 3) 2 = ((2) 6) 2 = (2) 12.
9. 1) () 2 (2);

2) (3) () 3.

22
1. 1) ) 3 (4) = 3+4 = 7; ) 3 () 4 = 3+4 = 7;
) () 3 4 = 3+4 = 7;
) () 3 () 4 = 3 4 = 3+4 = 7;
2) ) (2) 5 5 = 25+5 = 15; ) (2 5) 2 = (2+5) 2 = 14;
) (4) 4 4 = 4+44 = 20; ) ( 7) 7 = (7+1) 7 = 56;
3) ) (4) 2 (2) 4 = 42+24 = 16;
) ( 2) 2 ( 2) 3 = (2+1) 2+ (2+1) 3 = 15;
) (5) 2 (2 3) 2 = 52+ (2+3) 2 = 20;
4) ) 12 : (6) 2 = 1262 = 0 = 1; ) (4) 5 : (4) 2 = 4542 = 12;
) ( 2) 3 : ( 3) 2 = (1+2) 3 (1+3) 2 = 1 = .
2. ) ( 2) 5 : (2) 2 = 12;
) (() 3 () 4 ) 3 = 24.

) ((3 4 7) 2) 2 : (13) 2 = 30;

3. 1) ) 37 (32) 3 : 310 = 37+2310 = 33 = 27;

) 520 : (52) 5 : 58 = 520258 = 52 = 25;

(3 )

2 4

2) )
)

= 387 = 31 = 3 ;

37

(9 3)2 9 4
9

93 94

3) ) 1012 : (26 56) =

94

(2 )
(2 )

3 5

2 6

= 2 35 26 = 2 3 = 8 ;

= 9 3 = 729 ;

512 212
2 6 56

= 5 6 2 6 = 10 6 = 1000000 ;

1516

= 15 2 = 225 ;
1514
) 126 : (35 45) = 126 : 125 = 121 = 12.
) 516 316 : 1514 =

4. 1) ( 4) 2 : 8 = 2;
2) (3) 2 18 = 24;

3) (3) 2 (18) = 24;

4) 6 ( 2) 2 = (8) (4).

5. ,
, ,
, , , 00 .

35

 23
1. 1) 3,5 16 = 56; 3,5 0,04 = 0,14; 3,5 0 = 0; 3,5 1 = 3,4;
3,5 (10) 2 = 350;
2) 4 (729) = 2916; 0,5; 0; 108; 4000;
3) 28; 14;
4) 4; 32;
5) 4; 300.
2. 1)

22

-1
2

-0,8 -0,6 -0,4 -0,2

1,28 0,72 0,32 0,08

0
0

0,2 0,4 0,6 0,8

0,08 0,32 0,72 1,28

-10 -8
-6
-4
-2
0,13 -100 -51,2 -21,6 -6,4 -0,8

0
0

2
0,8

1
2

2)

3. 1) 0,6624;
2) 4147,2;

4
6,4

6
8
10
21,6 51,2 100

3) 0,3168;
4) 366,7356.

4. 1) = 0; = 2,5; = 2,5; = 25;

2) b = 2, c = 1; b = 5, c = 2;
1
b = 11, c = 0; b = 1, c = .
2
5. 1)
2)
3)
4)

, = 0 702 = 0;
, 0,042 0, .. 0,042 = (0,2) 2 0;
, = 0 252 = 0;
, = 1 63 = 6 < 0.

24
1. 1) ) 1,5 8 = 122;

2
a 12ab 2 = 8a 2 b 2 ;
3
) 0,52 () = 0,532;

2) )

) 2 43 = 45;
1

) 6 y y 2 = 2 y 3 ;

) 0,442 2,524 = 66.

2. 1) 104 (0,15) = 64;

104 (0,528) = 5312;
1
2) a 2 bc 15ab 2 c = 5a 3 b 3 c 2 ;
3
1 2
1
a bc 0,2abc 2 = a 3 b 2 c 3 .
3
15

3. 1) 62 4ab = 24a3b;

36

2) (62) 523 = 3035.

1
1
3) ) ab = a 3 b 3 ;
2
8

4. 1) ) (8) 2 = 642;
3

1 6
1
) a 2 =
a ;
3
27

) (0,23) 4 = 0,001612;
3

3 3

) (10a3b2) 4 = 10000a12b8;
) (2z3) 5 = x5y10z15;
4) ) (22) 2 = 424;
) (43) 3 = 6493;
) (2b3c4) 4 = a8b12c16.

2) ) (4) = 64 ;
) (82b) 2 = 64a4b2;
) (223) 3 = 869;
2

1 4 1 2
x = x ;
4
2
0,366b8 = (0,6a3b4) 2;
2) 0,001x6 = (0,1x2) 3; 125a3c9 = (5ac3) 3;

5. 1)

6. 1) ) 203 (5) 2 = 203 252 = 5005;

) 0,45 (23) 4 = 0,45 1612 = 6,417;
) (3) 2 126 = 6 126 = 1212;
4
1

2) ) 3 x 6 y 3 xy 2 = 81x 24 y 12 xy 2 = x 25 y 14 ;
81

81

16
8 3 15
2

) ab 5 18a 5 b =
a b 18a 5 b = a 8 b16 .
3
27
3

7. 1) ) (42) 3 (0,53) 2 = 6436 0,2562 = 1698;

3

2
8 6 9
2

x y 81x 8 = 24 x14 y 9 ;
) x 2 y 3 9 x 4 =
27
3

2) ) (24) 4 (64) 2 = 816 3682 = 361618;

) (10a3b2) 5 (0,2ab2) 5 = 100000a15b10 (0,00032a5b10) =
= 32a20b20.

8. 1) ) (92) 2 = 8124;
) , ,
10048 = (1024) 2 0;
2) ) 88 = (44) 2;
) 2733 276 = 27293 ,
(9 3 2
).

9 3
: 27 2 x 2

, , ,

37

 25
1. 1) ) 2 = 2 + 2;
) 3 62 52 7 = 182 352;
) 2 2 3b + a 8c = 6a3b + 8ac;
) 8 3 (5) 72 (4) = 1002 + 282;
2) ) 115 85 + 35 + 5 = 75;
) 1,93 2,93 3 = 23;
) 20 + 5 17 = 8;
) 8ab2 3ab2 + ab2 7ab2 = ab2.
2. 1) ) 3t2 5t2 11t 3t2 + 5t + 11 = 5t2 6t + 11, n = 2 ();
) 2 + 5 4 3 52 + 4 + 13 = 3 42 + 9 17, n = 3;
) m3 + m2 + m + 1 m4 m3 m2 m 1 = m4, n = 4;
2) ) 22 + 7 52 11 + 32 = 32 4 + 32, n = 2;
) 4b2 + a2 + 6ab 11b2 6ab = 7b2 + a2, n = 2;
) 3a2x + 3ax2 + 5a3 + 3ax2 8a2x 10a3 = 5a35a2x + 6ax2, n=3.
3. 1) 3 4 + 2 = 4, 15 + 4 4 = 15;
2) 2pq 2p p + 2q = 2pq 3p + 2q, 42 + 9 14 = 37;
3) 3uv3 + u2v2 2uv3 + u3v u4 = uv3 + u2v + u3v u4,
1 + 1 1 1 = 2.
4. 1) 4b3 + 5b2 3b + 15;
2) 43 + 52 + 3 + 15;

3) 108b3 + 45b2 9b + 15;

4) 1086 + 454 92 + 15.

5. ) 22 + 3pq q2 + 7q2 2qp + 5q2 9p2 pq 12q2;

) 27a2bc + 23ab2c 25abc2 11abc2 33a2bc + 48ab2c =
= 6a2bc + 71ab2c 36abc2.
6. ) 4 + 23 2 + 1 + 5 = 5 + 4 + 23 2 + 1;
) 6 35 + 5 + () 6 = 35 + 5;
) 35 + 2 11 + 11 = 35 + 2;
) a3b2 + ab2 + a2b4 + (a2b4) = a3b2 + ab2.
7. ) 3 11 5 + 17 8 + 23 + 10 =29;
) 32 53 + 42 + 82 5 + 11 + (112) = 53 + 42 + 11 5;
) 22 + 3 92 + 82 5 + 82 + 32 + 2 + 2 = 132.
8. 1) : 4 + 22 + 5, .. 4 = (2) 2 0, 22 0;
2) : 2 + u2 + 5, .. 2 0, u2 0;
: 2 u2 a4u2 3, .. 2 0, u2 0,
a4u2 = (a2u) 2 0.
C 26
1. 1) ) 72 5 + 3 + 72 5 = 142 5 2;
72 5 + 3 72 + 5 = 5 + 8;
) 3 + 1 32 3 + 1 = 32 + 2;
3 + 1 + 32 + 3 1 = 32 + 6;

38

) + 3b + 3a 3b = 4a;
a + 3b 3a + 3b = 2a + 6b;
) 2 5ab b2 + a2 + b2 = 2a2 5ab;
a2 5ab + b2 a2 b2 = 5ab 2b2;
2) ) 22 + 8 11 + 32 6 + 3 = 52 + 2 8;
22 + 8 11 32 + 6 3 = 2 + 14 14;
) 93 3 + 92 + 4 = 93 + 92 7;
93 3 92 + 4 = 93 92 2 + 1;
) 4m4 + 4m2 13 + 4m4 4m2 + 13 = 8m4;
4m4 + 4m2 + 13 4m4 + 4m2 13 = 8m2 26;
) 22 + 3pq + 8q2 + 6p2 pq 8q2 = 8p2 + 2pq;
22 + 3pq + 8q2 6p2 + pq + 8q2 = 4p2 + 4pq +16q2.
2. ) (2 + 5b) + (8a 11b) + (9b 5a) = 2a + 5b + 8a 11b + 9b
5a = 5a + 3b;
) (3x + 10y) (6x +3y) + (6y 8x) = 3x + 10y 6x 3y + 6y 8x =
= 11x + 13y;
) (82 + 3) + (72 11 +3) (32 4) = 82 +3 72 11 +
+ 3 + 32 + 4 = 42 8 + 7;
) (v + n k) (v u) + (v u + k) = v + u k v + u + v u + k = v + u.
3. 1 ();
2 + 5 ();
3 + 5 + 5 = + 10 ();
4 + 10 + 5 = + 15 ().

1)
2)
3)
4)
5)

+ 5 ();
+ 10 ();
+ + 5 = 2 + 5 ();
+ 10 + + 15 = 2 + 25 ();
+ + 5 + + 10 + + 15 = 4 + 30 ().

4. 1) 15m7 3m4 + m3 + 5 15m7 + 3m4 m3 5 = 0;

15m7 3m4 + m3 + 5 + 15m7 3m4 + m3 + 5 = 30m7 6m4 + 2m3 + 10;
2) 83 + 3a2b 5ab2 + b3 + 18a3 3a2b 5ab2 + 2b3 =
= 26a3 10ab2 + 3b3;
83 + 3a2b 5ab2 + b3 18a3 + 3a2b + 5ab2 2b3 = 10a3 + 6a2b b3.
5. 1) (3 5 8v) (2x + 7y 3v) + (5v 11x + y) =
= 3x 5y 8v 2x 7y + 3v + 5v 11x + y = 10x 11y;
2) (2a3 + 32 + 1) (44 + 63 22 + 2) (25 + 34 3 + 2)=
= 23 + 32 + 1 44 63 + 22 2 25 34 + 3 2 =
= 25 74 33 + 42 3 + 1.

39

6.

1
px
1)
3 + 5
5 16
2)
7x + 3
x2 18
3
2
3
3)
a + 3a b + b
0
4)
2x2y 3xy2 8
2x2y + 3xy2 + 8
5)
x2 + 2xy + y2
4xy
6)
3x + 2a
x 2a + b
p1 + px = p2, px = p2 p1, px .

2
8x 11
x2 + 7x 15
a3 + 3a2b + b3
0
x2 2xy + y2
2x + b

27
1. 1) ) + + + = ( + ) + ( + );
) 3 + 2 + 8 + = (3 + 2 + 8) + ;
2) ) 2 + + + 1 = (2 + ) + ( + 1);
) aq2 q aq + q2 = (aq2 aq) + (q2 q).
2. 1) ) bm bn m n = (bm bn) (m + n);
) bx + by + x y = (bx + by) (y x);
) ab + ac b c = (ab b) (c ac);
2) ) bx by b x + y + 1 = (bx by b) (x y 1);
) bx + by + x y b + 1 = (bx + by b) (x + y 1);
) a2 + b2 + 2a 1 = (b2) (a2 2a + 1).
3. ) ax + by c d = (ax + by) (c + d)
) 3x 3y + z a = (3x + z) (3y + a)

) 5x 3y z = 5x (3y + z);
) 2x + y z = y (2x + z).

4. ) (22 3 + b) (a2 5x + 1) (b + x2 7x) = 2x2 3a + b a2 +

+5x + 1 b x2 + 7x = (x2 + 12x) (2 + 3 1);
) (82 + 3ab2 b) (x2 ax2 b) x = 8ax2 + 3ab2 b x2 + ax2 +
+ b x = (9ax2 x2 x) + 3ab2.
C 28
1. 1) ) m (n + k) = mn + mk;
) k (a b + 2) = ka kb + 2k;
) l (q r) = lq + lr;
) ( t + 3) = xp + xt 3x;
2) ) 32 ( 3) = 33 92;
) 54 (2 3) = 105 + 57;
) 43 (2 ) = 45 + 43; ) (q10 q11) 8q15 = 8q25 8q26;
3) ) 3 (4 + 2 1) = 35 + 33 3;
) 5 (2 3 4) = 53 + 152 + 20;
) (4b2 4b + 16) 0,5b = 2b3 2b2 + 8b;
) 2 (22 8ab + b2) = 4a3 16a2b + 2ab2;
) 2 (5 3 + 2 1) = 7 5 + 23 2;
) 3z (5z3 + 2z2 z + 1) = 15z4 6z3 + 3z2 3z.
2. 1) ) m (n + k) = mn + mk; ) (q + r) (l) = lq lr;
2) ) (b + c m) a = ab + ac am;
) ab (c m + k) = abc + abm abk;

40

3) ) a2 (ab b2) = a3b a2b2; ) (a b) a2b2 = a3b a2b2. 3.

1)
) 3 ( + 1) + ( + 1) = 3 + 3 + + 1 = 4 + 4;
) 2 2 ( 2) = 2 2 + 4 = + 2;
) 3 ( + 5) 2 ( 6) = 3 + 15 2 + 12 = + 27;
) 13 (6b 1) 6 (13b 1) = 78b 13 78b + 6 = 7;
2) ) 3 ( 2) 5 ( + 3) = 32 6 52 15 = 22 21;
) 2 ( ) + (3 2) = 2 22 + 32 2 = 2;
) 2a (a b) + 2b (a + b) = 2a2 2ab + 2ab + 2b2 = 2a2 + 2b2;
) 3 (8 + 1) 8 (3 5) = 24 + 3 24 + 40 = 3 + 40;
3) ) m (m2 m) + (m2 m + 1) = m3 m2 + m2 m + 1 = m3 m + 1;
) 5n2 (3n + 1) 2n (5n2 3) = 15n3 + 5n2 10n3 + 6n =
= 5n3 + 5n2 + 6n;
) (2 2) + (2 2) = 3 2 + 2 3 = 3 3;
) (3 + 2 + ) (3 + 2 + ) = 4 + 3 + 2 3 2 = 4 .
4. 1) 2a (a + b) b (2a b) b (b + 1) = 2a2 + 2ab 2ab + b2 b2
b = 2a2 b;
2 (0,3) 2 + 0,4 = 0,58;
2) 2 (2 3 + 1) 2 (3 32 + ) + 4 33 + 2 =
= 4 33 + 2 24 + 63 22 + 4 33 + 2 = 0,

x = 1

1
0.
3

5. 1) ) 5 (4 + 5 26 + 37) = 55 + 66 77 + 88 =
= 88 77 + 66 + 55;
) (2x3 + 3x2 a a2) xya = 2x4ya + 3x3ya xya3 xya2;
2) ) 2x (5x3 3x bx + b3) b = 10x4b bx2b 2x2b2 + 2xb4 =
= 10x4b + 2xb4 2x2b2 6x2b;
) xt (x2t2 xt 3) p = x3t3p + x2t2p + 3xtp.
C 29
1. 1) ) (3 + 5) + (8 + 1) = 17;
3 + 5 + 8 + 1 = 17;
11 = 11;
= 1;
) 19 5 (3 1) = 9;
19 15 + 5 = 9;
15 = 15;

= 1;
2) ) 30 + 5 (3 1) = 35 25;
30 + 15 5 = 35 25;
20 = 50;

) (3 5,8) (2,2 + 3) = 16;

3 5,8 2,2 3 = 16;
8 = 16;
= 2;
) 21 = 20 8 (2 0,5);
21 = 20 16 + 4;
16 = 37;

x=

37
5
= 2 ;
16
16

) 10 (3 4) + 51 = 7 (5 + 3);
30 + 40 + 51 = 35 + 21;
5 = 0;
41

 = 2,5;
) 10 5 = 6 (8 + 3) 5;
10 5 = 48 + 18 5;
33 = 23;

x=

23
;
33

3) ) 6 (8 + 5) = 0;
48 + 30 = 0;
48 = 30;

= 0;
) 6 5 (3 + 2) = 5 ( 1) 8;
6 15 10 = 5 5 8;
14 = 3;

x=

3
;
14

) 8 (2 0,5) = 0;
16 + 4 = 0;
16 = 4;

5
x= ;
8

= 0,25;

) 6 (8 + 5) = 6;
48 + 30 = 6;
48 = 36;
= 0,75;

) 8 (2 0,5) = 8;
16 + 4 = 8;
16 = 12;
= 0,75.

2. 1) 8 7 = 0;
7 = 8;

x=

8
;
7

2) 0,2 1 = 3 0,8;
= 4;

3) 8 + 5 = 3 + 10 + 25;
5 = 30;
= 6;
4) 2 ( 4) + 8 = 8;
2 8 + 8 = 8;
6 = 0; = 0.

3. 1) ) 3 (1 2) 5 (3 ) 6 (3 4) = 83;
3 6 15 + 5 18 + 24 = 83; 19 = 71;

x=

71
14
= 3 ;
19
19

) 23 3 (b + 1) + 5 (6b 7) 7 (3b 1) = 0;
23 3b 3 + 30b 35 21b + 7 = 0;
4
1
6b = 8; b = = 1 ;
3
3
) (2 + 3) 5 (2 3) = 3 (7 );
22 + 3 52 + 15 = 21 32; 3 = 0; = 0;
2) ) 2m + m (3 (m + 1)) = m (2 m) + 12;
2m + 3m m2 m = 2m m2 + 12;
2m = 12; m = 6;
) 7 + 3 (k 3 (k + 5)) = 5 (7 2k) + k;
7 3k 9k 45 = 35 10k + k; 3k = 73;

k=

73
1
= 24 .
3
3

4. 1 () = 2 6; 2 () = 12 ;

42

1 (6) = 6 = 2 (6); 1 (9) = 12; 2 (9) = 3 .

30
1. 1) )

2x + 1
= 1;
5

2 + 1 = 5;
2 = 4;
= 2;
)

11 3x 1
= ;
4
2

11 3 = 2;
3 = 9;
= 3;

3x 8
= 1 ;
2

3 8 = 2;
3 = 6;
= 2;
2) )

3x + 7 6 x + 4
=
;
5
5

3 + 7 = 6 + 4;
3 = 3;
= 1;
)

2x 1 6 x
=
;
6
8

4 (2 1) = 3 (6 );
8 4 = 18 3;
11 = 22;
= 2;

7 x 3 5x + 1
=
;
6
2

7 3 = 3 (5 + 1);

3
;
4
2x + 3 4x 3
= 1;
3) )
+
3
3
8 = 6; x =

2 + 3 + 4 3 = 3;
6 = 3; = 0,5;
) x

10 x 1 4 x + 1
=
;
6
6

6 10 1 = 4 + 1;
8 = 2;
= 0,5;
)

x x+2 1
+
= ;
5
15
3

3 + + 2 = 5;
4 = 3;

x x3

= 1 ;
4
5

5 4 ( 3) = 20;
= 32;
)

2 x + 1 3x + 1
= 2;
+
5
7

7 (2 + 1) + 5 (3 + 1) = 70
14 + 7 + 15 + 5 = 70;
29 = 58;
= 2;
)

8 x 3 3x + 1
=2;

7
10

10 (8 3) 7 (3 + 1) = 140;
80 30 21 7 = 140;
43

x=

3
;
4

59 = 177;
= 3.

2 x 3 7 x 13 5 2 x
2. 1)
= x 1;
+
+
3
6
2
2 (2 3) + 7 13 + 3 (5 2) = 6 ( 1);
4 6 + 7 13 + 15 6 = 6 6; = 2;
2)

x 2 2x 5 4x 1
+
+
= 4x;
5
4
20
4 ( 2) + 5 (2 5) + 4 1 = 20 (4 );
38 = 114; = 3;

3) x 2 3x 1

2 x 2 + 3x 5
= 1,5 ;
2

22 6 2 22 3 + 5 = 3; 9 = 0; = 0.
31
1. 1) 3 + 7 + 5 11 = 12; 8 = 16;
2) 3 + 7 = 5 11 + 15; 2 = 3;
3) 3 + 7 = 5 11 + 15; 2 = 3;

4) 3 + 7 = 2 (5 11) ; 7 = 29;
5) 2 (3 + 7) = 5 11 + 6; = 19.

2. 1) ; + 8 ;
6 + 8 ( + 8) = 232;
14 = 168;
= 12 ;
2) ;

S
x
x
=
+1 ; t =
;
Vcp
20 60
3 = + 60;
2 = 60;
= 30 ();
3) ;
+ 10 ;
+ 10 + 12 = + 22 ;
9 + 18 ( + 10) + 9 ( + 22) = 1458;
36 = 1080;
= 30 (2) ;
30 + 10 = 40 (2) ;
30 + 22 = 52 (2) ;
4) V ;
V + 20 ;
44

3 (V + 20) + 2,5V = 280, .. 0,5 2,5

3V + 60 + 2,5V = 280;
5,5V = 220;
V = 40 (/) ;
40 + 20 = 60 (/) ;
5) ;
1 :
+ 3 ;
.. , :
+ 2 ( + 3) = 51;
3 = 45;
= 15 () ;
2 :
3 , :
= 2 ( 3) = 51;
3 = 57;
= 19 () .
32
1. 1) ) (2 + 3) = 2 + 3;
) (3 5) = 3 5;
) (7 + 1) = 7 + ;
) ( + 1) = ;
2) ) 5a (b + 2a) = 5ab + 10a2;
) 7n (2mn 1) = 14mn2 7n;
) 20c (c + 4b) = 20c2 + 80bc;
) 3 (2 + 4) = 32 122;
3) ) 3 ( + 1) = 4 + 3;
) 2z3 (z2 2) = 2z5 4z3;
) 6 (3 + 7 82) = 36 + 77 88;
) 52 (1 2 32) = 52 103 154;
4) ) ( + 3) = 2 + 3;
) (2 + 5 3) = 3 + 522 32;
) 3a2b (a 2b) = 3a3b 6a2b2;
) 222 (3 2 + 1) = 623 432 + 222.
2. 1) ) ( + ) ( + b) = x (a + c a b) = x (c b);
) y (2a + 3b) y (3a b) = y (2a + 3b 3a + b) = y (4b a);
) 2 ( + 2) + (3 ) = (2 + 4 + 3 ) = (5 + 3);
) 2 (3 7) 2 (5 + 3) = 2 (3 7 5 3)=2 (2 10) =
= 2 (2 + 10);
2) ) ( + ) + ( + ) = ( + ) ( + );
) (3 + ) z (3a + c) = (3a + c) (x z);
) (2 + 3) 3 (2 + 3) = (2 + 3) ( 3);
) 2k (3k 4) + (3k 4) = (3k 4) (2k + 1);

45

3) ) a (b c) + c (c b) = (b c) (a c);
) 2x (m n) (n m) = (m n) (2x + 1);
) 3 ( ) ( ) = ( ) (3 + );
) (b c) + a (c b) = (b c) (1 a).
3. 13.
2r r. :

S = 2ra +

r 2 r 2
+
= 2ra + r 2 = r (2d + r ) .
2
2

13. ,
2r
r. :

S = (2r )
2

r 2
2

r 2
2

= 4r 2 r 2 = r 2 (4 ) .

4. 1) ) 7a4b3 14a3b4 + 21a2b5 = 7a2b3 (a2 2ab + 3b2);

) 833 + 8823 1634 = 823 ( + 11 2);
) 22b2c2 4a2bc2 + 2a3c = 2a2c (b2c2 2bc + a);
2) ) (a + 3) (b + 5) (a + 3) (b + 6) = (a + 3) (b + 3 b 6) =
= (a + 3) = 1 (a + 3);
) (3 1) (8b + 1) + (7b 3) (1 3x) = (3x 1) (8b + 1 7b+3)=
= (3x 1) (b + 4);
) (3 + 10) (6 5) (8 9) (5 6) = (6 5) (3 + 10 +
+ 8 9) = (6 5) (11 + 1);
5.
2 3 1 = 11;
2 3 = 12;
1) 3 (2 3 1) = 3 11 = 33;
2) (2 3 1) (2 3) = 11 12 = 132;
3) 8 (2 3) 9 = 8 12 9 = 87.
33
1. 1) ) ( + 3) (b 7) = ab 7a + 3b 21;
) (a 5) (11 b) = 11a ab 55 + 5b;
) (8 a) (b + 2) = 8b 16 ab 2a;
) (7 b) (a 7) = 7a + 49 ab + 7b;
2) ) ( 4) ( + b) = x2 + 8x 4x 32 = x2 + 4x 32;
) ( 5) (9 ) = 9 2 45 + 5 = 2 + 14 45;
) (3 + ) (1 ) = 3 3 2 = 2 4 3;
) ( 10) ( 6) = 2 6 + 10 + 60 = 2 + 4 + 60;
3) ) (8 + 3) (2 1) = 16 8 + 6 3;
) (2 1) (3 + 7) = 62 + 14 3 7 = 62 + 11 7;
) (3 2b) (2a 3b) = 6a2 9ab 4ab + 6b2 = 6a2 13ab + 6b2;
) (15 + 27) (5 9) = 752 135 135 254 =
= 752 270 243;

46

4) ) (32 1) (2 + 1) = 63 + 32 2 1;
) (32 1) (22 + 1) = 64 + 32 22 1 = 64 + 2 1;
) (m2 n) (m + n2) = m3 + m2n2 mn n3;
) (m2 n) (m n2) = m3 m2n2 mn + n3;
5) ) ( + 2) (2 3)=3 2 3 + 22 2 6=3 + 2 5 6;
) (5b 1) (b2 5b + 1) = 5b3 25b2 + 5b b2 + 5b 1 =
= 5b3 26b2 + 10b 1;
) (m n + 1) (m + n) = m2 + mn mn n2 + m + n =
= m2 n2 + m + n;
) (m 2n) (m + 2n 1) = m2 + 2mn m 2mn 4n2 + 2n =
= m2 4n2 m + 2n;
6) ) 2 (b + 1) (b + 3) = 2b2 + 6b + 2b + 6 = 2b2 + 8b + 6;
) 8 ( 1) ( + 5) = 82 40 + 8 + 40 = 82 32 + 40;
) b (3b + 1) (2b 5) = 6b3 15b2 + 2b2 5b = 6b3 13b2 5b;
) 5m (m n) (m + 3n) = 5m3 + 15m2n 5m2n 15mn2 =
= 5m3 + 10m2n 15mn2.
2. 1) ) (m2 m 1) (m2 + m + 1) = m4 + m3 + m2 m3 m2 m m2
m 1 = m4 m2 2m 1;
) (3n2 + 2n + 1) (3n2 + 2n 1) = 9n4 6n3 + 3n2 + 6n3 + 4n2
2n + 3n2 + 2n 1 = 9n4 + 10n2 1;
2) ) ( + 1) (4 3 + 2 + 1) = 5 4 + 3 2 + + 4 3 +
+ 2 + 1 = 5 + 1;
) (2 + 3 + 5) ( 1) = 2 2 + 2 4 + 3 + 6 5 =
= 6 5 4 + 3 + 2 + 2;
3) ) ( + 3) ( 5) (2 + 2 15) = (2 2 15) (2 + 2 15) =
= 4 + 23 152 23 42 + 30 152 30 + 225 =
= 4 342 + 225;
) ( + 1) (2 + 1) (6 3 + 1) = (3 2 + + 2 + 1) (6
3 + 1) = (3 + 1) (6 3 + 1) = 9 6 + 3 + 6 3 + 1 = 9 + 1.
3. (5 10) (3 7) = (10 5) (7 3) = 5 (2 ) (7 3).
4. ) (m 1) (m + 4) = m2 + 3m 4

) ( + 3) ( 2) = 2 + 6.

34
1. 1) ) (3 + 5) (3 6) + 30 = 92 18 + 15 30 + 30 = 92 3;
) 3b2 + (8 3b) (b + 5) = 3b2 + 8b + 40 3b2 15b = 7b + 40;
2) ) 8 (3 + 1) (5 + 1) = 8 152 3 5 1 = 152 1;
) 8 (3 + 8) (2 5) = 8 62 + 15 16 + 40 =
= 62 + 7 + 40;
3) ) ( 3) ( + 5) (2 + ) = 2 + 5 3 15 2 = 15;
) ( + 2) ( + 3) ( 1) = 2 + 3 + 2 + 6 2 + = 6 + 6;
) ( 3) + ( + 1) ( + 4) = 2 3 + 2 + 4 + + 4 =
= 22 + 2 + 4;

47

) ( + 2) ( + 3) ( 3) = 2 + 2 2 + 3 3 + 9 = 2 + 9.
2. ) (3 + 5) (4 1) = (6 3) (2 + 7);

122 + 17 5 = 122 + 36 21; 19 = 16; x =

16
;
19

) (5 1) (2 ) = ( 3) (2 5);
10 52 2 + = 2 52 6 + 15; 6 = 4; x =

2
.
3

3. ) ( + ) (2 + 2) ( 2) = 2 + 2 3 + 22 2 + 23 =
= 3 + 32 + 23;
) (5 7) (7 + 5) (7 5) (5 + 7) = 352 + 25 49
352 352 49 + 25 + 352 = 48;
) (3 + 2) (2 2) (2 + 2) (3 2) = 5 23 + 22 42
5 + 22 23 + 42 = 43 + 42.
4. 1) 20t + 25 (t 2) = 45t 50; 2) t + 1 + t 2 = 2t 1;

3) Vcp =

45t 50 45t 50
S
; Vcp =
.
=
t
t +t 2
2t 2

5. 1- ; + 10 ; + 10 2-
; + 10 + 10 = + 20 ;
: V = abc, a, b, c , , , V ;
25 ( + 10) ( + 20) = 25 ( + 10) + 20000;
20 = 20 3 = 20 103 3 = 20000 3;
252 + 750 + 5000 = 252 + 250 + 20000; 500 = 15000;
= 30 () 1 () ;
30 + 10 = 40 () .
35
1. 1) ) a (b + c) + p (b + c) = (b + c) (a + p);
) a (x y) b (x y) = (x y) (a b);
) 3a (a + b) m (a + b) = (a + b) (3a m);
) 7 ( ) + ( ) = ( ) (7 + );
2) ) ( 2) + ( 2) = ( 2) ( + 1);
) ( + 8) ( + 8) = ( + 8) (1 );
3) ) 2 ( 3) + b (3 a) = (a 3) (2 b);
) 3 (b 5) a (5 b) = (b 5) (a + 3);
4) ) ( 5) + (5 ) = ( 5) ( 1);
) m n + (n m) y = (m n) (1 y).
2. 1) ) (a + b) + c (a + b) = (a + b) (x + c);
) 3 ( ) + ( ) = ( ) ( + 3);
2) ) 4 (a + b) + y (a + b) = (a + b) (y + 4);
) 6 ( + 7) + ( + 7) = ( + 7) ( + 6);
3) ) ( + ) 5 ( + ) = ( + ) ( 5);
) a (b c) 4 (b c) = (b c) (a 4).

48

3. 1) ) 2a + b + 2a2 + ab = 2a (1 + a) + b (1 + a) = (a + 1) (2a + b);

) 3a + 3a2 b ab = 3a (1 + a) b (1 + a) = (a + b) (3a b);
) 22 3 + 4 6 = (2 3) + 2 (2 3)=(2 3) ( + 2);
) 22 + + + = ( + 1) + ( + 1) = ( + 1) ( + );
2) ) ab + ac + am + yb + yc + ym=b (a + y) + c (a + y) + m (a + y)=
= (a + y) (b + c + m);
) 22 + 33 + 22 = (1 + 22) (1 +
+ 22) = (1 + 22) ( );
3) ) bn + 1 + bn + b + 1 = bn (b + 1) + b + 1 = (b + 1) (bn + 1);
) am+2 1 a + am+1 = am+1 (a + 1) (a + 1) = (a + 1) (am+1 1).
4. ) 2 + + 2 + 2 = ( + 1) + 2 ( + 1) = ( + 1) ( + 2);
) 2 3 2 + 6 = ( 3) 2 ( 3) = ( 3) ( 2).
36
1. 1) a2 + b2;
2) (a b) 2;
3) p3 q3;
4) (m + n) (m n);
5) x2 + 2xy.
2.

2 + 2
(2) 2 + () 2

( + ) 2
(2 + 3) 2

92 2
(3) 2 b2
62 (5b) 2

(9 a) 2
(3a b) 2

3.
(ax) 2 + 112
a2 + 12

(t + 4y) 2
(m + 12) 2

(9b) 2 12
02 a2

(6 x) 2
(11 11x) 2

4. ) (a + b) 2 + (a b) 2;
) 2 (2 + 2) (2 2).
37
1. 1) ) ( + 4) 2 = 2 + 8 + 16;
) (9 + ) 2 = 81 + 18 + 2;
) ( + ) 2 = 2 + 2 + 2;
2) ) ( 7) 2 = 2 14+ 49;
) (8 b) 2 = 64 16b + b2;
) (11 ) 2 = 121 22 + 2;
3) ) (5 + 1) 2 = 252 + 10 + 1;
) (3 4) 2 = 92 24 + 16;
) (10 + 4) 2 = 100 + 80 + 162;
4) ) (2 3) 2 = 42 12 + 92;
) (5a + 6b) 2 = 25a2 + 60ab + 36b2;

49

) (3 + ) 2 = 92 6 + 2;
5) ) (2 9) 2 = 4 62 + 9;
) ( 3) 2 = 2 23 + 6;
) (2 + b2) 2 = a4 + 2a2b2 + b4.
2.

25a2 + 10ab + b2

25a2 10ab + b2

3a

1
b
3

5a

0,2b

25a2 + 2ab + 0,04b2

25a2 2ab + 0,04b2

ab

a2b2 + 8ab + 16

a2b2 8ab + 16

a2

2x

a4 + 4a2x2 + 42

4 42 + 42

22

36 + 1222 + 44

36 1222 + 44

9a 2 + 2ab +

1 2
b
9

9a 2 2ab +

1 2
b
9

3. 1) ((a + b) + c) 2 = (a + b) 2 + 2c (a + b) + c2 = a2 + 2ab + b2 + 2ac +

+ 2bc + c2;
2) ((a b) c)2=(a b)2 2c (a b) + c2=a2 2ab + b2 2ac + 2bc + c2
3) (x + y + z) 2 = x2 + y2 + z2 + 2xy + 2xz + 2yz;
4) (x y z) (x y z) = (x y z) 2 = x2 + y2 + z2 2zy 2xz + 2yz.
4. 2 (2 ) 2 = 0,5 (4 2) 2 = 0,5 (2 (2 )) 2 = 0,5 22 (2 ) 2
4 (2 ) 2 = (4 2) 2 = (2 (2 )) 2 = 22 (2 ) 2 = 4 (2 ) 2
38
1. 1) ) 2 + (3a b) 2 = a2 + 9a2 6ab + b2 = 10a2 6ab + b2;
) 9b2 (a 3b) 2 = 9b2 a2 + 6ab 9b2 = a2 + 6ab;
) (5a + 7b) 2 70ab = 25a2 + 70ab + 49b2 70ab = 25a2 + 49b2;
) (8a b) 2 64a2 = 64a2 16ab + b2 64a2 = b2 16ab;
2) ) (5 + ) 2 + ( 7) = 25 + 10 + 2 + 2 7 = 22 + 3 + 25;
) (4 ) + (4 ) 2 = 4 2 + 16 8 + 2 = 16 4;
) ( 8) 2 2 (6 ) 2 = 2 16 + 64 72 + 242 23 =
= 23 + 252 88 + 64;
) ( + 7) (1 ) 2 = 2 + 7 1 + 2 2 = 9 1;
3) ) 2 ( b) 2 = 2a2 4ab + 2b2;
) (1 + 2) 2 = + 42 + 43;
) 6 (2 ) 2 = 242 + 24 62;
) (3 ) 2 = 92 + 62 3.
2. 1) ) (a 3b)2 + (3a+b)2=a2 6ab + 9b2 + 9a2 + 6ab + b2=10a2+10b2;
) ( + 2) 2 ( 2) 2 = 2 + 4 + 42 2 + 4 42 = 8;

50

2) ) (( (a2 + b2) 2 2a2b2) 2 2a4b4) 2a8b8 = ((a4 + b4) 2 2a4b4)

2a8b8 = (a8 + b8) 2a8b8 = a8 2a8b8 + b8;
, , ..
(( ((a + b) 2 2ab) 2 2a2b2) 2 2a4b4) 2 2a8b8,
. , .
, : = (a8 + b8) 2 2a8b8 = a16 + b16.
3. 1) (3a + 4b) 2 + (3a 2b) 8b = 9a2 + 24ab + 16b2 + 24ab 16b2 =
= 9a2 + 48ab = 3a (3a + 16b);
2) (6 2) 2 (5 + 2) 2 = 362 24 + 4 252 20 4 =
= 112 44 = 11 ( 4).
4. ;
( + 3) 2 = 2 + 39 2 + 6 + 9 = 2 + 39; 6 = 30; = 5.
39
1. 1) ) 42 + 4ab + b2 = (2a + b) 2;

) 4a2 4ab + b2 = (2a b) 2;

2

2) )

9 2
16
4
3
a 2ab + b 2 = a b ;
16
9
4
3

1 2
1

a + ab + b 2 = a + b ;
4
2

) b2 2a2b + a4 = (b a2) 2.
3) ) a2b2 + 2ab + 1 = (ab + 1) 2
)

2. ) 92 + 6ab + b2;
) 252 10ab + b2;
) 4 4b + b2;
) 36a2 + 24ab + 4b2;
4a2 + 24ab + 36b2;
9a2 + 24ab + 16b2;
144a2 + 24ab + b2.
3. ) 16a2 + 8ab + b2 = (4a + b) 2; 36a2 + 12ab + b2 = (6a + b) 2;
2

4
2

36a + 8ab + b 2 = 6a + b ;
9
3

1 2
1

)
m + mn + 4n 2 = m + 2n ;
16

4
49m2 + 28mn + 4n2 = (7m + 2n) 2;

49m 2 + mn +

1 2
1
n = 7m + n .
196
14

C 40
1. 1) ) ( + 2) ( 2) = 2 4;

3) ) ( + 2b) (a 2b) = a2 4b2;

51

) (3 ) (3 + ) = 9 2;
) ( ) ( + ) = 2 2;

) (3 ) (3 + ) = 92 2;
) (5 + 2) (5 2) = 252 42;

4) ) (4a b) (b + 4a) = 16a2 b2

2) ) (3b 1) (3b + 1) = 9b2 1;
2
) (5b + 6) (5b 6) = 25b 36; ) ( + 7) (7 ) = 49 2;
1
1
1
) (7 a)(7 + a ) = 49 a 2 ; ) (4b + 1) (1 4b) = 1 16b2.
2
2
4
2.

3
2x
0,3

b
3
4b

1
p
3

1
c
6

ab
2

5
2

(3a + b) (3a b)
(2 + 3) (2 3)
(0,3a + 4b) (0,3a 4b)
1
1 1
1
( p + c)( p c )
6
3
6 3
(ab + 5) (ab 5)
(2 + 2) (2 2)

9a2 b2
42 92
0,09a2 16b2

1 2 1 2
p
c
9
36
a2b2 25
4 4

3. 1) ) (8a + b) (b 8a) = b2 64a2; ) (8a b) (8a + b) = 64a2 b2;

) (8a b) (b + 8a) = b2 64a2;
2) ) (5 + 22) (5 22) = 252 44;
) (2a + 3b3) (3b3 2a) = 9b6 4a2;
) (a2b3 + 1) (1 a2b3) = 1 a4b6;
3) ) (xn 2) (xn + 2) = x2n 4; ) (a2n + b) (a2n b) = a4n b2n;
) (an+1 bn1) (an+1 + bn1) = a2n+2 b2n2;
4) ) (( + ) ) (( + ) + ) = ( + ) 2 2;
) (a b + 4) (a b 4) = (a b) 2 16;
) (a2 b2) (a2 + b2) (a4 + b4) (a8 + b8) = (a4 b4) (a4 + b4) (a8 + b8) =
= (a8 b8) (a8 + b8) = a16 b16.
41
2

1
1
1
1
1

1. ) (5a + b)(5a b) = 25a 2 b 2 ) 3 x + = 9 x 2 + 2 x + ;

3
3
9
3
9

) (ab cx) (ab + cx) = a2b2 c2x2;

2

1
1

y 2 x = y 2 2 xy + 4 x 2 ;
4
2

) (0,4 10) (0,4 + 10) = 0,162 1002;

) ( 3) 2 = 22 6 + 9.
2. 1) ) (2a b) (2a + b) + b2 = 4a2 b2 + b2 = 4a2;
) ( + 7) 2 10 = 2 + 14 + 49 10 = 2 + 4 + 49;
) 92 ( + 3) ( 3) = 92 2 + 92 = 182 2;

52

) 5b2 (a 2b) 2 = 5b2 a2 + 4ab 4b2 = b2 + 4ab a2;

2) ) ( ) ( + ) ( 2) 2 = 2 2 2 + 4 42 = 4 52;
) ( + 3) 2 ( 3) 2 = 2 + 6 + 9 2 + 6 9 = 12;
) ( + 3) 2 + (b + 3c) (b 3c) = a2 + 6ac + 9c2 + b2 9c2 =
= a2 + 6ac + b2;
) ( 4) 2 + ( + 4) 2 = 2 8 + 162 + 2 + 8 + 162=22 + 322
) ( 3) ( + 3) ( + 8) ( 8) = 2 9 2 + 64 = 55;
) (2 + 1) (2 1) + ( 7) ( + 7) = 42 1 + 2 49 = 52 50.
3. ) (2a + 2b) (a b) = 2 (a + b) (a b) = 2a2 2b2;
) ( ) (5 + 5) = ( ) 5 ( + ) = 52 + 52;
) (4 + 4) ( + ) = 4 ( + ) ( ) = 42 + 8 + 42;
) (3 3) (7 7) = 3 ( ) 7 ( ) = 21 ( )2=212 42 + 212.
4. ) (3 + 1) (3 1) + (5 + 1) 2 = 92 1 + 252 + 10 + 1 = 342 + 10 =
= 2 (17 + 5);
) (3p 2k) (2k + 3p) (3p k) 2 = 9p2 4k2 9p2 + 6pk k =
= 6pk 5k2 = k (6p 5k).
5. 1) (22 1) (22 + 1) (24 + 1) (28 + 1) 216 = (24 1) (24 + 1) (28 + 1)
216 = (28 1) (28 + 1) 216 = 216 1 216 = 1;
2) (2 + 1) (22 + 1) (24 + 1) (28 + 1) (216 + 1) 232=(2 1) (2 + 1) (22 +
+ 1) (24 + 1) (28 + 1) (216 + 1) 232 = 232 1 232 = 1.
42
1. 1) ) 42 1 = (2 1) (2 + 1); ) 1 92 = (1 3) (1 + 3);
) 25 162 = (5 4) (5 + 4);
2) ) m2 a2 = (m a) (m + a);
) n2 + b2 = (b n) (b + n);
) 4x2 q2 = (2x q) (2x + q);
3) ) 2 92 = ( 3) ( + 3);
) 812 2 = (9 ) (9 + );
) 362 2 = (6 ) (6 + );
4) ) 492 1212 = (7 11) (7 + 11);
) 1002 25b2 = (10a 5b) (10a + 5b);
) 1442 16k2 = (12y 4k) (12y + 4k);
5) ) 22 1 = ( 1) ( + 1);
) 2 2b2 = (c ab) (c + ab);
) 24 9 = (2 3) (2 + 3).
2. 1) ) 25 3622 = (5 6) (5 + 6);
) 1004b2c2 121 = (10a2bc 11) (10a2bc + 11);
2) ) (3 + 1) 2 (4 + 3) 2 = (3 + 1 4 3) (3 + 1 + 4 + 3) =
= ( 2) (7 + 4);
) (a + b + c) 2 (a b c) 2 = (a + b + c a + b + c) (a + b + c +
+ a b c) = (2b + 2c) 2a;

53

3) ) x2n 9 = (xn 3) (xn + 3);

) x2n y2n = (xn yn) (xn + yn);
2
4n
2n
2n
) k a = (k a ) (k + a ); ) 81a4n 1 = (9a2n 1) (9a2n + 1);
4) ) 2 (5 + 10) + (2 8) (3 + 2)=102 + 20 + 62 + 4 24
16 = 162 16;
) (3 + 5) (4 5)2 (2,5 + 1,5)=(3 + 5) (4 5) (5 + 3)=
= (3 + 5) (4 5 ) = (3 + 5) (3 5).
3. (n + 1) 2 n2 = (n + 1 n) (n + 1 + n) = 2n + 1 = n + (n + 1);
(n, n + 1 ).
43
1. 1) ) (4a b) (a 6b) + a (25b 3a) = 4a2 24ab ab + 6b2 +
+ 25ab 3a2 = a2 + 6b2;
) (2 + 3) ( ) ( + ) = 22 2 + 3 32 2 =
= 2 32;
) 3 ( + 1) + ( + 2) ( 3) = 32 + 3 + 2 3 + 2 6 =
= 42 + 2 6;
) 2 (5 3) ( 2) ( 4)=102 6 2 + 4+ 2 8=92 8;
2) ) (3a + b) (a 2b) + (2a + b) (a 5b) = 3a2 6ab + ab 2b2 +
+ 2a2 10ab + ab 5b2 = 5a2 14ab 7b2;
) ( + 1) ( + 7) ( + 2) ( + 3) = 2 + 7 + + 7 2 3 2
6 = 3 + 1;
) ( 4) ( + 6) + ( 10) ( 2) = 2 + 6 4 24 + 2 2
10 + 20 = 22 10 4;
) ( 3) (5 ) (4 ) ( + 6) = 5 2 15 + 3 4 24 +
+ 2 + 6 = 10 39.
2. 1) ) 3 (3 + 7) (3 + 1) 2 = 92 + 21 92 6 1 = 15 1;
) 4b (3b + 6) (3b 5) (3b + 5) = 12b2 + 24b 9b2 15b +
+ 15b + 25 = 3b2 + 24b + 25;
2) ) ( 2) ( + 3) ( 1) 2 = 2 + 3 2 6 2 + 2 1 = 3 7;
) ( 5) ( 1) ( 6) 2 = 2 5 + 5 2 + 12 36 = 6 31;
3) ) ( + 1) 2 ( + 2) 2 = 2 + 2 + 1 2 4 4 = 2 3;
) ( 4) 2 (4 ) (4 + ) = 2 8 + 16 16 + 2 = 22 8;
4) ) 4 ( + 5) 2 (42 + 40) = 42 + 40 + 100 42 40 = 100;
) (4ab b2) + 2 (a b) 2 = 4ab b2 + 2a2 4ab + 2b2 = 2a2 + b2.
3. ) (7 ) (7 + ) + ( + 3) 2 = 49 2 + 2 + 6 + 9 = 6 + 58;
6 (3,5) + 58 = 37;
) (2a b) 2 (2a + b) 2 = (2a b 2a b) (2a b + 2a + b) =

= 2b (4a) = 8ab; 8

10 7

= 8 ;
7 10

4. 1) ) 3 (2a 5b) 2 12 (a b) 2 = 12a2 60ab + 75b2 12a2 +

+ 24ab 12b2 = 63b2 36ab;
) 7 (2 + 5) 2 + 5 (2 7) 2 = 282 + 140 + 175 + 202 140 +

54

+ 245 = 482 + 420;

2) ) (32 + 4) 2 + (32 4) 2 2 (5 32) (5 + 32) = 94 + 242 +
+ 16 + 94 242 + 16 50 + 184 = 364 18;
) (43 + 5) 2 + (43 1) 2 2 (43 + 5) (43 1) = (43 + 5
(43 1)) 2 = (6) 2 = 36. (2 2ab + b2 = (a b) 2);
3) ) ( 2) ( + 2) ( ) (2 + + 2) = 2 42 3 2
2 + 2 + 2 + 3 = 3 3 + 2 42;
) (2 1) 2 2 ( + 1) (2 + 1) = 43 42 + 23 + 22
2 22 + 2 2 = 23 42 + 2.
5. 1) (2a b) (2a + b) + (b c) (b + c) + (c 2a) (c + 2a) = 0;
4a2 b2 + b2 c2 + c2 4a2 = 0;
2) (3 + ) 2 (3 ) 2 = (3 + 1) 2 (3 1) 2;
(3 + 3 + ) (3 + + 3 ) = (3 + 1 3 + 1) (3 + 1 +
+ 3 1);
2 (6) = 2 (6); 12 = 12 .
44
1. 1) ) 32 12 = 3 (2 4) = 3 ( 2) ( + 2);
) bx2 9b = b (x2 9) = b (x 3) (x + 3);
) 50b 2a2b = 2b (24 a2) = 2b (5 a) (5 + a);
) 22 2 = 2 (2 1) = 2 ( 1) ( + 1);
2) ) 22 982 = 2 (2 492) = 2 ( 7) ( + 7);
) 33 + 3ab2 = 3a (b2 a2) = 3a (b a) (b + a);
) 22 23 = 2 (2 2) = 2 ( ) ( + );
) 3 3 = (2 2) = ( ) ( + ).
2. 1) ) 3a2 6ab + 3b2 = 3 (a2 2ab + b2) = 3 (a b) (a b)=3 (a b)2;
) 2 + 4 + 4 = (2 + 4 + 4) = ( + 2) 2 = ( + 2) ( 2);
) a2b 4abc + 4bc2 = b (a2 4ac + 4c2) = b (a 2c) 2 =
= b (a 2c) (a 2c);
) 22 4 + 2 = 2 (2 2 + 1) = 2 ( 1) 2 = 2 ( 1) ( 1);
2) ) 5a2 10ab 5b2 = 5 (a2 + 2ab + b2) = 5 (a + b) 2;
) 32 + 12 12 = 3 (2 4 + 4) = 3 ( 2) 2;
) a2 + 10ab 25b2 = (a2 10ab + 25b2) = (a 5b) 2;
) 123 122 3 = 3 (42 + 4 + 1) = 3 (2 + 1) 2.

1 2
1
1
1
1
2
a ab + b 2 = (a b ) ) a 3 + 3 = (a + 3) a 2 3a + 9 ;
2
2
2
9
9
2) ) 6 6 = (3) 2 (3) 2 = (3 3) (3 + 3) = ( ) (2 + +
+ 2) ( + ) (2 + 2);
) 5 23 + = (4 22 + 1) = (2 1) 2 = ( 1) 2 ( + 1) 2;
3) ) 2 ( 3) 2 ( 3) + ( 3) = ( 3) (2 2 + 1) =
= ( 3) ( 1) 2;
) 1 2 4 (1 2) + 42 (1 2) = (1 2) (1 4 + 42) =
= (1 ) (1 + ) (2 1) 2;

3. 1) )

55

4) ) a3 + 8b3 + a2 2ab + 4b2= ( + 2b)(a2 2ab + 4b2) + (a2 2ab +

+ 4b2) = (a2 2ab + 4b2)(a + 2b + 1);
) a3 + 8b3 + a2 + 4ab + 4b2 = (a + 2b) (a2 2ab + 4b2) + (a + 2b)2=
= (a + 2b) (a2 2ab + 4b2 + a + 2b).
4. 1) ( 1) 3 4 ( 1) = ( 1) ( + 1) ( 3) = ( 1) (2 2 +
+ 1 4) = ( 1) (2 2 3) = ( 1) (2 3 + 3) =
= ( 1) ( ( 3) + 3) = ( 1) ( 3) ( + 1);
2) (2 + 1) 2 42 = ( 1) 2 ( + 1) 2 = (2 + 1 2) (2 + 1 + 2) =
= ( 1) 2 ( + 1) 2;
5. 1) ( + 1) ( + 2) = 2 + 3 + 2; 2) (2 + 3 + 2) ( + 1) = 3 + 42 + 5 + 2.
45
1. 1) )

=6

=2

6
(5; 6)

= 2 - 4

1
0

(2; 2)

=4-

-4

2) )

=-1

y=

3
1

1
0
-1

(1; 0)
1

-6
1

0
-2

= 3 - 3
(-6; -6)

56

-6

2
x2
3

2. . 14;
(2; 3);

. 14;
(2; 1);

y = 0,5 x + 2 ;
y = 5 x

y = x +1
y = 4 1,5 x ;

0,5 + 2 = 5 ;
1,5 = 3;
= 2;
0,5 2 + 2 = 3 = ;
(2; 3);

4 1,5 = + 1;
2,5 = 5;
= 2;
= 2 + 1 = 1;
(2; 1).

3. 1) )

= 2
6

=6

4
2

(1; 1)

(2; 4)

1
1

=2

2) )

(-2; 2)
2
1
-2

y=

1
x +1
2

= -

57

= 2 + 1

1
-1
(-1; -1)

-1
-2

= - 2

4. )
y=

1
x+3
2
3

2
1
0

1 2

-4

(4,7; 0,7)

=-4

y=
1
0

1
x2
2

-2
(2,5; -0,8)
= 3 1,5

)
58

y=

1
x 1,5
2

1
0
-1,5

(2,2; -0,4)

= 4 2

y = 3 x 5 ; : k (; 3) (3; +);
5. 1)

y = kx + 4

k 3; : k = 3;
: k .

y = 1,5 x 1 ; : k ;
2)

y = 1,5 x + k

: k 1;
: k = 1;
;
: ;
: ;

3) y = 0,5 2 x ;
y = 0,5 1,5 x

: k 3;
: k ;
: k = 3.
C 46
1. 1) ) = 5 ; = 5 ; ) = ; = ;
) = 3;
2) ) = 3 6;
3) ) = 1,5;
2
1
y = x+2;
x= y;
3
3
) = 3 + 2;
) = 5 2,5;
2
1
3
x = 2 y ;
x= y ;
5
2
2

59

) = 5;

) x =

1
y= x;
5

y=

7
y 1,4 ;
4

4
x 0,8 .
7

y = 5 x
y = 4 4 + 1 = 5
2. 1) )
3x + 5 x = 7 ; x = 1 ; 3 1 + 4 = 7 ;

y = 3 ; 3 ( 3) = 0 ;
;

x 3 x = 6 x = 3 3 3( 3) = 6
y = x3
y = 1 1 4 = 3
)
2 x + x 3 = 9 ; x = 4 ; 2 4 + 1 = 9 ;

=
2
+
3
=
7
y
x
y

2 2 + 7 = 3 ;
;
)

3x 2 x 3 = 1 x = 2 3 2 7 = 1
x= y
)

45 6n 2n = 5 ; n = 5 3 5 2 5 = 5 ;
m = 5 5 + 2 5 = 15

a = 2 3b
a = 5 5 + 3 ( 1) = 2
)
4 6b + 3b = 7 b = 1 2 5 + 3 ( 1) = 7 ;

3
6
5
14

=
=

1 3 3 5 ( 1) = 14 ;
p
p
p
;
)
k = 1 2 p
k = 3 ; 3 + 2 ( 1) = 1

=
2

2
=
0
2

0=2 .
d
c
d

;
;
)
3c 4c + 4 = 3 c = 1 3 2 0 = 3

2) )
m = 15 2n

x + y = 10 x = 10 y
x = 6
3. )
x y = 2 ; 10 y y = 2 ; y = 4 ;

x + y = 74
)
x y = 16 ;

x = 45
x = 74 y
74 y y = 16 ; y = 29 .

x + y = 1 2 x
4. 1) z = 2 x
;
x y = 3
x = z y
2) y + z = 4
;
z + 2 y = 6

y = 1 3 x
x y = 3 ;
z = 2 x

y = 1 3x

x 1 + 3 x = 3;
z = 2x

x = 2 y
y + 4 + y = 4 ;
z + 2 y = 6

x = 1

y = 2 ;
z = 2

x = 2
y = 0 .
z = 4

47

3x 3 y = 21 ; 5 x = 39
1. 1) )

2 x + 3 y = 18 y = x 7

60

2a 2b = 4 ; 3a = 1 ;
)

5a + 2b = 3

b = 2 a

 3 p + 9q = 15 ; 11q = 11 ;
)

3 p + 2q = 4
p = 5 + 3q
6a + 4b = 6 19b = 57
2) )
6a + 15b = 63 ; 3a 2b = 3 ;

18

81
=

180
77 y = 154
x
y

;
)
2 x + 9 y = 20 ;
18 x + 4 y = 26

12 x + 10 x = 4 ; 4 x = 26
)

12 z 6 x = 30

4 z 2 x = 10

2 x = 12 ; x = 6 ;
2. 1) )

x + y = 5 y = 1

4n = 16
n = 4
)
;
.
2
n
+
m
=
5

m = 3
3u 3v = 12 8v = 8
2) )
;
;
3u 5v = 20
u + v = 4

2a = 4 a = 2 ;
)
a b = 1 ;

b = 3

v = 1
;

u = 5

21x 7 y = 35 23 x = 46 x = 2
;
;
;
)
2 x + 7 y = 11 3 x y = 5 y = 1
4 m 5n = 1
n = 3
)
;
;

4
m
+
6
n
=

2m 3n = 2
6 x + 9 y = 3 y = 1
;
3) )
;
6 x 10 y = 4 2 x + 3 y = 1
6n 9d = 3
)
;
6n 8d = 48

n = 3
;

m = 3,5
y = 1
;

x = 1

17 d = 51
;

2n 3d = 1

d = 3
;

n = 4

4a + 6b = 0
25a = 75 a = 3
;
.
)
;
21a 6b = 75 2a + 3b = 0 b = 2
x 1 + y 1 = 6
x + y = b
4 x = 20 x = 5
3. 1)
;
;
;
;
3 x 3 y + 1 = 10 3x y = 12 x + y = 8 y = 3
10a + 5 + 14b + 14 = 7 10a + 14b = 12
2)
;
;
6a 4 + b + 4 = 16
6a + b = 16
7

a=3
10a + 224 84a = 12
37
.
;

b
=
16

6
a

b = 3 5
37

61

x + y + z = 1

4. 1) x y = 2 ;
x + z = 5

x + y z = 4

2) x y z = 0 ;
2 x = 6

x y = 2

x + z = 5 ;
y = 4

x = 2

y = 4
z = 7

x = 3

x y z = 0 ;
2 x 2 z = 4

x = 3

y = 2 .
z = 1

48
x 7 y = 0
1. 1) )
;
12 x + y = 17

x = 7 y
x = 1,4
;
;

84
y
+
y
=
17

y = 0,2
y = 5x 1
y = 5 x 1 y = 1,5
)
;
;
;
x
+
15
x

3
=
5

16 x = 8
x = 0,5

9 x + 2 y = 16 x = 2
9 x + 2 y = 16
2) )
;
;
;

9
x
+
15
y
=

33

17 y = 17
y = 1
12 x + 20 y = 8
12 x + 20 y = 8 x = 16
)
;
;
;

12
x

21
y
=

18

y = 10
y = 10

3 x + 2 y = 0
3) )
;
3 x + 4 y = 1,5

3 x + 2 y = 0
;

6 y = 1,5

x =
6 ;

y = 0,25

2 x 6 y = 18 2 x + 30 + 18 x = 18 x = 0,6
)
;
;
.
3 x + y = 5 y = 5 3 x
y = 3,2

y = 3x + 6 y = 3x + 6
y = 1,8
2. )
;
;
; (1,4; 1,8);
y = 2 x 1 2 x 1 = 3 x + 6 x = 1,4
4 x + 3 y = 8
)
;
3 x 2 y = 6

8 x + 6 y = 16
;

9 x 6 y = 18

2
x = 1 + 0,5 y
3. ) 5
;
2 x = y 2
5

x = 2
17 x = 34
;
; (2; 0).

3
x

2
y
=
6
y = 0

2
x = 0
x = 1 + 0,5 y
;
;
5
1 + 0,5 y = y 2 y = 2

8a + 5b = 14 8a + 5b = 14
b = 2
b = 2
;
;
;
.
)
4a + 3b = 8 8a 6b = 16 4a + 3b = 8 a = 0,5

62

a = 5 b
4.
;
15 3b + 2b = 1
a + 2b = 11
1)
;
a 2b = 1

a = 9
;

b = 14

x=

9;

y = 1

14

2a = 10
;

a + 2b = 11

a = 5
;

b = 3

x = 5
;

y = 1

5a 6b = 2
10a + 12b = 4 3b = 9
b = 3
2)
;
;
;

10a 9b = 13 10a 9b = 13
5a 6b = 2 a = 4
3a = 9 a = 3
a b = 1
a b = 1
;
;
;
3)
2a 0,5b = 5 4a + b = 10 a b = 1 b = 2

x = 4
;

y = 1

x =
3 ;

y = 0,5

49

x + y = 17
1. 1) )
;
x + 7 = y

x + y = 17
;

x y = 7

x y = 12
)
;
x = 4 y

x = y + 3 x y = 3
2 x + 2 y = 400 2 x + 2 y = 400
;
2) )
;
; )
;
x + y = 36 x + y = 36
x = 3 y
x 3 y = 0
4 x + 5 y = 730
3) )
;
x = y + 70

4 x + 5 y = 730
;

x y = 70

3 x + 2 y = 580
)
.
5 x + y = 780

2. 1) 26,
5;
2) 2 . 54 .

a + b
x + y + z = 16
2 = 22,5
x y = 215

3. 1)
; 2)
; 3) x = 0,25 y
.

2
a
b
0
,
8
x

0
,
6
y
=
129

x = z 4

=1

3
3
50
1). ;
;

63

 x + y = 1500
;

x y = 300

x = 900( )
2 x = 1800
;
;

x
+
y
=
1500
y = 600( )

2). 5 .; 20 .;
x = 7 ( p.)
x + y = 10
x = 10 y
;
;
;

5 x + 20 y = 95 50 5 y + 20 y = 95 y = 3 ( p.)
3). ;
96 x + 24 y = 528 96 x 24 y = 528 48 y = 480
;

96 x = 24 y + 48
96 x = 24 y + 48 96 x 24 y = 48

y = 10 ( )
;

x = 3 ( )

4). 1 ; 2- ;
2 x + 3 y = 330 2 x + 3x + 30 = 330 x = 60 ( / )
;
;
;

y = x + 10
y = x + 10
y = 70 ( / )
5). ;
x + 3 x + 4 = 44 x = 10 ( )
x + y = 44
;

3(x + 2 ) = y + 2 y = 3 x + 4
y = 34 ( )
6. ;
;
x = 150 ( )
x + y = 270
x = 270 y
;
;
;

0
,
6
x

0
,
7
y
=
6
162

0
,
5
y

0
,
7
y
=
6

y = 120 ()
7. ;
;
2(x + y ) + 3(x y ) = 36
5 x y = 36

;
;
2

x 5 y = 0
x y = 3 (x + y )

25 y y = 36 y = 1,5
;

x = 5 y
x = 7,5
S = (2 + 3) 7,5 = 37,5 .
51
1. ) 1)

64

1
1
1
9

1
9
= = 1,125 ;
8
8

9

) 1)

0,21
= 0,21 ;
1

2.

2)

0
=0;
0 1

3)

6
=2;
4 1

2)

1,5

4
x +1

1
3

3 x
x +1

1
3

3. ) 0;
) 3 0; 3;

) 5 + 1 0; c

1
3 2
+
2 3

1
6
=
;
13 13

6

1
3

1
3

) ;
) b , .. b2 + 9 > 0.
1
;
5

4. ) ( 4) ( + 4) = 0, = 4, = 4,
, , 4 4;
) ( 2) ( + 11) = 0; = 2 = 11,
, 2 11.
52
1. 1) )

b
2
b
b
; ) ; ) ; ) ;
y
3
c
c

2) )

a b
;
c

ab
;
a+b

a(a b )
a
=
;
(a b )(a + b ) a + b
a(a + b )
a
=
;
(a b )(a + b ) a b

x 1
x
x +1
x
; )
; )
; )
;
x 1
x
x +1
x
1
1
4) )
; )
; ) ; ) m 2n;
m
a + 3b

3) )

5) )
)

(a b )2
a b

= a b;

(a + 2b )2
a + 2b

= a + 2b ;

0,7 + 1
1,7
=
= 1 ;
2,7 + 1 1,7

(a + 2b )2 = a + 2b ;
(a 2b )(a + 2b ) a 2b
65

2a(3a p )
= a ;
2(3a p )
2a(3a p )
)
= a;
2(3a p )

6) )

2. )

(a 3c )(a + 3c ) = a 3c ;

(a 3c )(a + 3c ) = a 3c .

3c + a

3c a

(37 23)(37 + 23) = 14 60 = 14 = 7 ;

(47 13)(47 + 13) 34 60 34 17
(45 + 13)2 = 58 2 = 58 .
58

58

x(a + b ) + a + b (a + b )(x + 1) x + 1
=
=
, a + b 0;
y (a + b ) + a + b (a + b )( y + 1) y + 1
0,7 + 1
1,7
=
= 1 ;
2,7 + 1 1,7
a + b + (a + b )(a b ) (a + b )(1 + a b ) a + b
)
;
=
=
2
(a b )(1 + a b ) a b
a b + (a b )

3. )

1 + b 0;

1,75 + 1,76
3,51
=
= 351 .
1,75 1,76 0,01

53

9
;
x
9
3
)
= ;
3x x
a2
2) )
;
a3
a2 9
)
= a + 3;
a 3

1. 1) )

5 x 1 + 3x + 1 8 x 8
= 2 = ;
x
x2
x
8 x 2 + 3 + 5 x 2 3 13x 2
)
=
= 13x ;
x
x
a 2 6a + 9
)
= a 3;
a3
a2 4
a+2
)
.
=
b(a 2)
b

15
;
a
2 7a + 11a 5 57 + 6a 10a 60
=
= 10 ;
2)
a6
a6
a +1
5a + 9 5a 8
1
3)
=
=
.
2
(a 1)(a + 1) a 1
a 1

2. 1)

66

(3a 1) = 3a 1 ;
9a 2 6a + 1
=
2
(3a 1)(3a + 1) 3a + 1
9a 1
2

3. 1) )

)
2) )

5a 3 + 3a 1 + 5 4a 3 3a 12

(a + 2)2

x2

(x 3)(x 2)

a3 8

(a + 2)2

1
;
x3

a 2 4a + 4 (2 a )
=
= 2a ;
2a
2a
2
4a 2 + 4a + 1 (2a + 1)
)
=
= 2a + 1 .
1 + 2a
2a + 1
2

54

x 2 + xy xy
x2
4b + b 5
5b 5
=
; )
;
=
b(b 5)
b(b 5)
y (x + y )
y (x + y )
3a 3b + 5a + 5b 8a + 2b
= 2
;
)
a2 b2
a b2
5c
c 2 2c c 2 3c
)
=
;
(c + 3)(c 2)
(c + 3)(c 2)

1. 1) )

a 2 + 2ax + x 2 a 2 + 2ax x 2
4ax
= 2
;
2
2
a x
a x2
y2 + 4y + 3 y2 4y 4
1
)
;
=
( y + 2)( y + 1)
( y + 2)( y + 1)

2 + 3x 3 y
;
x2 y2

18 + 35
53
=
;
15(x 2 ) 15(x 2 )

a 2 + ab a 2
ab
= 2
;
a2 b2
a b2

4b 4a 4b
4
=
;
a(a + b )
a+b

59
2
=
;
6(m + 1)
3(m + 1)

x 2 + xy x 2
xy
;
=
3(x + y )
3(x + y )

2) )

1 + a 2 3a + 2 a 2 3a + 3
x 2 + 4x + 2
=
;
)
;
x+4
a2
a2
2
3a + 3c a + 2c 2a + 5c
2 xy + x 2 + y 2 (x + y )
)
=
; )
;
=
xy
xy
a+c
a+c

3) )

67

b 2 9 1 b 2 10
=
;
b+3
b+3

a 2 + b 2 2ab (a b )
.
=
ab
ab
2

a + 3 a + 2b 2b + 3
;
=
ab
ab
2b 3 4b + 2b 2 + 12b 2b 2 + 10b 3
)
;
=
16b 2
16b 2
6 3 x 5 x 10 + 2 x 5 6 x 9 6 x + 9
2) )
;
=
= 2
(2 x )(2 + x )
4 x2
x 4

2. 1) )

)
3. 1) )

n 2 + n 2 + 2nx + x 2 x 2
2nx
.
= 2
x2 n2
x n2

(a + b)(a 2 ab + b 2 ) + (a b)(a 2 + ab + b 2 ) =

a2 b2
a +b a b
2a 3
;
= 2
=
(a + b )(a b )
a b2
3

(a b )(a 2 + ab + b 2 ) (a + b )(a 2 ab + b 2 ) =
(a + b )(a b )

a3 b3 a3 b3
2b 3
2b 3
;
= 2
= 2
2
(a + b )(a b )
a b
b a2

2) )

x 2 + 2 x + 1 x 2 3 + 4 x 2 10 x + 6 4 x 2 8 x + 4
=
=
2 x2 1
2 x2 1

4(x 1)
2(x 1)
;
=
2(x 1)(x + 1)
x +1
2

=
)

2x 2 + 4x + 2x 2 4x 2x 2 + x 2 + 4
2 x(x 2)(x + 2)

3x 2 + 4
2 x 3 8x

x +1 x +1
1
1
2
;

=
= 2
x 1 x +1
x2 1
x 1
x + 39
6
5
6 x + 24 5 x + 15
)
;

=
= 2
(x 3)(x + 4)
x3 x+4
x + x 12

4. 1) )

3x
6 x 1 6 x 2 + 3x 6 x 2 + 13x 2
16 x 2

=
= 2
=
2
x 2 2x + 1
2 x 3x 2
2 x 3x 2
ax + b
a = 16
= 2
;
b = 2 ;

2 x 3x 2
a
b
ax + a + bx b (a + b )x + a b
1
;
)
+
=
=
= 2
2
2
x 1 x +1
x 1
x 1
x 1

2) )

68

a+b = 0
:
a b = 1 ;

2 a = 1 ;
a + b = 0

a = 0,5 .
b = 0,5

C 55

1
5
; )
; ) 9;
21
2x 4 y
2
1
3
m
; ) ; )
; ) ;
2) )
p
3
p
q

1. 1) )

3) )
4) )
5) )
6) )
)

3a 11 10b14 2a
5a 6c
8a 2 b 8c

= 1 ; ) 15
=
;
)
2 = 64 ;
c
3c 10a
a b
5b
21a 10 7b
a
; ) 1; ) (a + b) ( );
c
(a + b ) = 3 (a + b ) ;
3
(3a 5b ) ; ) 5b ; ) 3(a 2b )
a b
2
2(a 2b ) 2
5a 3a + b 5(3a + b )
2a
; )

=
;
1 3a 2
3a
5(3a b )
(a b )(a + b ) = a b .
(a + b )(x + 3 y ) x + 3 y

2a 2 b
2
= 2 ;
2 3
3a b
3b

a 2 b 2 5a 5a 3 b 2 5a 2
;

=
=
3b 3a b
9
9ab 2
a 2 b 2 5a a 2 3ab
3a 3 b
a2
a 2 b 2 5a 5a 2
)
; )

=
: =

=
= 2 .
2
3
9ab b
9
3b 3a b 3b 5ab
15ab
5b

2. )

(a 3b )(a + 3b) (c 4d )(c + 4d ) = (a + 3b)(c 4d ) =

3b a
c + 4d
(c + 4d )2
(a + 3b)(4d c ) ;
=

3. 1) )

4d + c

2)

(a + b )(a b + 1) 2(x y ) = 2(a b + 1) ;

)
(x y )(x + y + 1) 3(a + b ) 3(x + y + 1)
2
4a 2 (2a b )(2a + b ) 3a(2a b ) 12a 3 (2a b ) (2a + b )
)

=
=
2a b
(2a b )12a 3 2a 2
12a 3
2a 2
(2a b )(2a + b ) ;
=
2a 2

69

x(x 1) (x + 1) (x 4 )(x + 4) (x + 1)(x 4 )

.

=
2(x + 1) x(x + 4)
3(x 1)
6
2

)
56

3a 2
3b 2
3(a b )(a + b )
a b 3ab

=

=
= 3a 3b ;
a+b
b a a+b a+b a+b
4ab
14ab 6b 2 4a 2 14ab
7 a 3b 2a 7b
)
+
=
+
=
2
2
2b 2a 3b
2a 2 3b 2
2a 2 3b 2
2a

1. 1) )

2 2a 2 3b 2
= 2;
2a 2 3b 2
x 2 a 2 2x 2
xa
2x x a
a+x
)
=
2
=

2
2
xa a + x
x( x a )
x + a2
x
=

a 2 + x 2 (x a )
1
= ;
x
x( x a ) x 2 + a 2

2a a 2 b
b 2 2ab + a 2
b
(a b ) b =
+ 2
2) ) 1
=

= 2
2

b
a
b
a
b
b
b
b (a b )

ab
;
=
b
a a b + a b a (b + a )b b + a

) 1 + : 1 =
=
=
;
:
b
b
b(b a ) b a
b b
2

3a 2 2a 1 a 1 + 3a 2

a
=
=
+ 1 : 1
:
a 1
a 1 1 a a 1
a 1
2a 1
2a 1
=
2
= 2
.
a 1 3a + a 1 3a + a 1

2. 1) ) 1

a + 3b
1
2a 2 6ab a + 3b a 3b

=
=
2a 2a(a 3b )
2a(a 3b )

2a 2 6ab 2a a 3b 1
=
;
2a(a 3b )
a 3b
a 3b
2
2
1 2a a 3b
a + 3b
=
=
=
;
) 1

2a a + 3b a + 3b a
a(a + 3b )
a(a + 3b )

)
70

1
1
1
2a 2 6ab a 2 + 9b 2 + 2a 2 + 6ab

+
=
=
a + 3b 2a a 3b
2a(a + 3b )(a 3b )

3a 2 + 9b 2
;
2a a 2 9b 2

8a + 2a 4a 2 + 1 4a 2 + 2a + 4a 2 + 4a + 1 4a 2 10a

=
2a(2a + 1)
8a 3 1
2

2) )

(4a

)(

+ 2a + 1 4a 2 4a + 1
(2a 1)
=
=
(2a 1) 4a 2 + 2a + 1 2a(2a + 1) 2a(2a 1)(2a + 1)
2a 1
=
;
2a(2a + 1)
=

=
3.

x 2 + 2x 1 + x 2 + 2x + 1 4x 2 1 1 + x 2 x 2 + x3
:
=
x2 1
x 2 (1 x )

4 x 2 + 4 x x 2 (1 x ) 4 x(1 x )(1 x ) 4(x 1)

=
=
.
(x 1)(x + 1)x
x +1
x2 1
x3

2
x
2x
2

x + 3p
=

x 3 p
x 3 p 3x x + 3 p 3x
x 3p
2x
2
2x
2x
x
2

=

+ 2
=

= 0.

3p
x 3 p 3x 3x
x
p
x
p
x

II
C1

1 4 3+ 4 7
2 5 85 3 1
+
=
=
=
= ;
; )
=
5 15
15
15
3 12
12
12 4
16
36 160 108 + 160 268
)
+
=
=
= 12 ;
21
7
21
21
21

1. 1) )

2) )
)
3) )
)
)

3 1 27 8 19
5 3 55 + 18
7
) + =
;
=
=
=1
8 9
72
72
6 11
66
66
33
262 7 1048 203 845
=
=
=7
;
116
29 4
116
116
7 31 21 62
41

=
=
;
30 45
90
90
41 249 82 249
167
7
;

=
=
= 4
20 40
40
40
40
22
5 83 25 249
224
112

=
=
=
= 2
.
45
18 30
90
90
45

2. 1) ) 13 + 40 + 27,13 + 50,07=53 + 77,2=130,2; ) 101,95+7,13=109,08

71

2) ) 5,47 3,009 = 2,461; ) 7,83 12 = 4,17.

3. 1) )

1 2
2
1 2 2
1 3
1
= ;
= ; )
= ; )
1 3 3
115
5
1 3
3

5 6
16 5
25 12
= 2 ; ) = 4 ; )

= 10 ;
3 5
5 4
6 5
5 21 3
16 9 12
2
1 32
= ; )

=
=2 ;
= 16 ; )
)
7 25 5
3 20 5
5
2 1
3
5 3
; ) = 3 ;
) 3; )
32
1 5
) 25,9; ) 20,35; ) 11,62;
) 720 : 8 = 90; ) 16,16 : 4 = 4,04; ) 28,9 : 17 = 1,7.
) 2197;
) (32 10) 2 = 322 102 = 102400;
) 1024;
) (13 100) 3 = 133 1003 = 2197000000;
) 343; ) 121; ) 0,343; ) 0,0121;
2
25
16
7
4
;
) =
)
=1 ;
49
9
9
3

2) )
3)
4)
5)
6)
4. 1)

2)
3)

1
;
27

4 4 4 64
10
4
) = =
.
=2
3 3 3 27
27
3

5. , ;
232 = 23 23; 3 3 = 9 ;
12322 = 1232 1232; 2 2 = 4 ;
1443; 4 4 4 = 64 ; 1313; 1 1 1 = 1 .
6. 1) ; x > 0. = 7 , = 7;

2) x x = x : 3 = x
7. 1)

1
1
; x= .
3
3

66...6 = 33...3 ;
33...3 123
33...3 100

33...3 = 266...67 ;
2)
6...6
123
98
26...667

3)

4)

x 222...2 = 88...8 ;
4 123
100
8.....88
x

22...2
5

11...110

= 11
...3
11 0 .
12
100

2
3
3
1. 1) ) 7 + 4 = 3 + 3 = 6 ;
7
4

72

62 31 4 31 4 4
) = = ;
5 31 5 31 5
5

57 29 8
57 8 49
7 6 9 4
9 37
14

=
=
= 7 ; )
= 2
=
=1 ;
7
7 29
7 7
7
3 7 4 23
23 23
23
2) ) 0,72 + 40 = 40,72
) 3,12 : 3,75 = 0,832;
) 4,24 1,364 = 2,876 ) 0,5 3 = 2,5.

2. 1) (5,8) 2 6,32 = 33,64 39,69 = 6,05;

2) (5,1 3,9) 2 = 81;
3

1
1
1
1
3) 5 5 = 0,253 = 3 = .
4
2
64
4

16 1 5
5
;
= =
5 1 16 16
273 51 22 100000 306306 100
16
2)

=
= 1612136
;
10 10 10
19
19
19
100 93 25
98 201 45 13 100
3)
= 29,4 20,1
=

=

24 10 6
15 45 10 6 52

3. 1) 1 : (4 0,8) = 1 : 3,2 = 1 :

279 125 154 77

2
=
=
=5 ;
30
30 15
15
8 125 13
9 45 132 8 1625
4)
=

= 0,54 + 1,8
+
22
33
100
13
10000
13
125 80

= 0,54 + 1,8 0,1 = 2,24.

=

4. ;
1) 1182 4; 33 7;
1182 33 14 7 = 7;
( , 14);
2) 173 3; 132 9;
173 132 13 9 = 4;
3) 1553 5; 652 5;
1553 652 5 5 = 0.
3

20
= 60 ;
100
20
2) 2
0,4 ;
100

1. 1) 300

2. 1) 13% 260;
100% ;
260 100
x=
= 2000 ;
13

20
= 0,9 ;
100
20
4) 0,05
= 0,01 .
100
3) 4,5

2) 13% 6,5;
100% ;
6,5 100
x=
= 50 ;
13
73

3) 0,0042 13%; 100%;

0,0042 100 42 1
42
14
7
;
x=
=
=
=
=
13
100 13 1300 650 325
169
4) 1,69 13%; 100%; x =
= 13 .
13
12
100% = 50% ;
24
12
2)
100% = 1% ;
1200

3. 1)

4. 1) ) 65% 0,65;
) 20% 0,2 ;
) 50% 0,5;
) 25% 0,25;
) 12,5% 0,125;

12
100% = 0,5% ;
2400
12
4)
100% = 10000% .
0,12
3)

2) ) 0,25 25%;
) 0,6 60%;
) 0,12 12%;
) 1,25 125%;
) 1,3 130%.

5. m 500 ; m ; m ; m ;
mo
= 0,8 ; m = 0,8 500 = 400 ;
1)
m
2) m = m mo = 500 400 = 100 ;
3) W = 100% 80% = 20% (.. 80% );
m
100
100% =
100% = 25% .
4)
400
mo

104,5 480
= 501,6 ;
100
2) 501,6 480 = 21,6 .

6. 1) 100% 480; 104,5% ; x =

7. 1)
2)
3)
4)

8. 1)

2)

40 + 0,2 40 = 48 ;
48 + 0,25 48 = 60 ;
60 40 = 20 20 ;
, :
+ 0,2 + 0,25 ( + 0,2) = + 0,2 + 0,25 + 0,05 = + 0,5 =
= + 50%. 50 .
40
40 2
0,1
0,13

100% = 2,5% ;
100% = 10000% .

9. ;
+ 0,5 0,5 ( + 0,5) = 0,25 = 25%. 25%.

74

4
13 19 65 57
2

=
= 8 ;
3 5
15
15
6 25
25
2) 21; 6; 0;
=
= 12,5 ;
1 12
2
3) 5; 7; 84,2; 7,6;
5 2 17
2
4) 11; 11,5; 4 + =
= 5 ; 2,25.
2 3 3
3

1. 1) 2,6; 0;

2.

3 5
5 3
(5 3)

3
14
14
6

2
11
11
4

1
8
8
2

0
5
5
0

1
2
2
2

2
1
1
4

3
4
4
6

9
12 = 14,25 ;
4
) 18 + 0 = 18; 2,8 + 14,7 = 11,9;
2) ) 24 + 24 11 = 11; 7 + 20 11 = 2;
) 1 54 + 63 = 10; 1 + 24 + 56 = 81;
3) ) (1,3 1,8) (1,3 + 1,8) = 3,1 0,5 = 1,55;
) 8 0,7 (16,5 16,5) = 8;
16,1 32,2 16,1 16,1
+
=

=0.
)
3
6
3
3

3. 1) ) 9 + 105 = 96;

4. S = V t; S = 25 1,2 = 30.
5. 1) a + b;

2) am1;

4)
5)

am1 + bm2;
am1 + bm 2
;
a+b

3) bm2.
1
1
8
; 8 = ;
11
11 11
1
1
1
2) : = 0; 5 0 = 0 ; : = 1; 5 ( 1) = 5 ;
3
3
3
3) : = 10; 0,7 10 + 3 = 10;
: = 2; 0,7 2 + 3 = 1,4 + 3 = 4,4;
4) : = 0,05; 2 0,05 0,1 = 0; : = 1; 2 1 0,1 = 1,9.

6. 1) : = 5; 8 5 = 40; : x =

7. 1) ( ) + z = 3 5 = 2;

2) 3 (5) 3 = 18;
3) (x y) 2z = 3 + 10 = 13;

4) 7 3 + 15 = 36;
5
5
= ;
5)
3+5
8
6) 3 + 15 = 18.
75

8. ) 6ab = 600 + 10a + b ; ) x7 y = 100 x + 70 + y ;

) 8 pp = 800 + 10 p + p = 800 + 11 p .
5

1 1 3 + 5 8 1 1 1+ 3 2
+ =
=
; + =
= ;
5 3
15
15 6 2
6
3
2 8
1 1 1 1
8 2 8 10
2
. + > +
=
= < 0 , , >
3 15
6 2 5 3
15 3
15
15
2 1 1 2 2 1 1
2 1 1
=
>0
) = +
17 3 17 3 3 3 17 17 3 17

1. 1) )

2 1
1 2
> ;
17 3
17 3

1 3
2) ) < 0 < 1 ;
8 4
1 4
1
1
1 4
) + 0,5 = + 0,8 0,5 = + 0,3 > 0 , + > 0,5 .
9 5
9
9
9 5

2. 1) = 0 : 3 + 1 = 1; x =

7
: 3x + 1 = 6
3

= 0 , x = 2

1
;
3

2) = 2 : 5 2 = 1; = 2 : 5 2 = 9;
= 2 , = 2;
3) = 0,7; = 0,9;
4 + 10 = 2,8 + 9 = 6,2;
= 1,4;
= 1,37;
4 + 10 = 5,6 13,7 = 8,1;
= 0,7 = 0,9 ,
= 1,4 = 1,37.
3. 1) ) t > 7; : t = 8, t = 9, t = 111; : t = 0, t = 7, t = 1;
) V 1,17; : V = 2, V = 3, V = 10;
: V = 1,16, V = 0, V = 4;
) 0; : = 0, = 2, = 4;
: = 1, = 21, = 1,01;
2) ) 8 > b 7; : b = 2, b = 0, b = 7;
: b = 8, b = 10, b = 8;
) 0,06 < a < 0,07; : = 0,065, = 0,067, = 0,0687;
: = 1, = 0, = 2;
) 0 < q 0,1; : q = 0,01, q = 0,05, q = 0,099, q = 0,1;

76

: q = 2, q = 0, q = 1.
4. w1, w2 ;
15
w1 =
100% = 75% ;
5 + 15
7
w2 =
100% = 70% ; w1 > w2;
7+3
.
5. 1) 3,5 0,24 3,5 = 3,5 (0,24 1) < 0, 3,5 0.24 < 3,5;
2) 3,5 0,24 0,24 = 0,24 (3,5 1) > 0, 3,5 0,24 > 0,24;
3) 3,5 0,24 (3,5) = 3,5 (0,24 + 1) > 0, 3,5 0,24 > 3,5;
1
1
4) 0,57 < 0,57 6 , 0,57 : 6 < 0,57 : ;
6
6
1
5) 0,57 : < 0,57 8 , 0,57 8 (0,57 ) = 0,57(8 + 1) < 0 ,
8
1
0,57 : < 0,57 ;
8
94 64
1
6)
+
=
(64 94 ) < 0 ,
2,1 2,1 2,1
94 : (2,1) < 64 : (2,1).
6. 1)

1
1
3
8
11
;
; ; ; ;0;
20
17 17 17 17

7. 1) 2,07; 2,007; 0; 1,65; 1,66;

2) (0,3) 3; (0,3) 2; 0,3.

2) (1,1) 2; 1,1; (1,1) 3.

8.
: 0,2
:
0,2 0,25 ( 0,2) = 0,2 0,25 + 0,05 ( 0,2) 0,2
: , 0,2
: 0,2, 0,2 0,2
,
( 0,2).
6
1. 1) ) (6,89 + 3,11) + (5,37 + 4,63) = 10 + 10 = 20
8 7
8
5
) 4 + 11 + 8 + 14 = 16 + 23 = 39
13 15
15
13
2) ) 921 321 + 457 = 600 + 457 = 1057
) 2,83 4,83 + 3,99 = 1,99
7
5
3) )
15 37,4 = 7 37,4 = 261,8 ) 36 2,7 = 10 2,7 = 27
15
18

77

9 17 13 5
= 1 1 = 1
17 9 5 13
11 37 5

) ( 28) = 1 10 = 10
37 11 14

4) )

1
5
= 40 + 1 = 41 ; ) 12 7 + 12 = 84 + 5 = 89
8
12
4
4

2) ) 9 7 + = 9 7 + 9 = 63 + 4 = 67
9
9

2. 1) ) 8 5 + 8

1
1

) 13 5 + = 13 5 + 13 = 65 + 1 = 66
13
13

3. )
4.

2 5 8 11 14 17
2
1
1 15 13 11 9 7 5 3 1

=
=
)
=
5 8 11 14 17 20 20 10
15 13 11 9 7 5 3 5 5

1 1 2 1 1 1 1 3 2 1 1 1 4 3 1
=
= ; =
=
; =
=
;
2 4
4
4 4 6
12
12 6 8
24
24
1 1 54 1
1 1 65 1
1 1 76 1
;
;
;

=
=

=
=

=
=
8 10
40
40 10 12
60
60 12 14
84
84
1 1
1
1
1
1 1 1 1 1 1 1 1
+ +
+
+
+
= + + +
4 12 24 40 60 84 2 4 4 6 6 8 8
1
1 1
1 1 1 1 7 1 3
+ +
=
=
=
10 10 12 12 14 2 14
14
7

5. 1) ) 888 25 = 222 4 25 = 222 100 = 22200

) 25 244 = 24 4 61 = 6100
) 2424 25 = 25 4 606 = 60600
2) ) 25 4 11,11 = 1111; ) 25 4 2,03 = 203
) 25 4 0,00909 = 0,0909
7
1. 1) ) 3,4 5 = 17
) 4,5 8 = 36
) 0,7 3,8 = 2,66

) 5 (12) = 60
2. 1) ) 5b + 7b = 12b

) 6 13 = 7

78

2) ) 6 1,25 = 7,5
) 7,5 2 = 15
) 0,6b 4y = 2,4by
5
11
1
)
p q = pq
11 20
4
) 0,2 = 1,2
1
) x 3 x = 2,8 x
5

) 5 8 = 13
) 3,4 + 8 = 4,6

1
1
7
a+ a =
a
3
4
12
5
2
) b b = b
7
7

) 5,4 + = 4,4
2) ) 9 + 17 30 + 4 = (9 + 17 30 + 4) = 0
) 5 + 11 + 47 31 = 22
) k k 5a a a a = 2k 8a
) 5,17 + 9,31 + 4,83 2 = 17,31
3) ) 15 + b 6b = 14a 5b
) 12 12 + 7 + 6 = 6 5
) 1,7 1,2 1,7 + 0,5 = 0,7
) 7 + + = 7
) 3 a b 12b 7 = 2a 13b 7
) 1,8 + 3 2,8 0,2 2 = 0,2 2,8 + 2,8
3. 1) ) k + (m + n) = k + m + n
) k (m n) = k m + n
) k (m + n) = k m n
) k (m n) = k + m + n
2) ) (x y) + (a + b) = x y + a + b
) (a c) (b d) = a c b + d
) a + (b c) (x y) = a + b c x + y
) 25 (m n) (a b) = 25 m + n a + b
4. 1) ) 2 + (3 8b) = 2a + 3a 8b = 5a 8b
) 5b (8a 5b) = 5b 8a + 5b = 8a
) 6 + (7 3) = 6 + 7 3 = 3 + 7
) (4 18) + 18 = 4 + 18 + 18 = 4 + 36
2) ) (5 2b) (7 + 10b) = 5 2b 7 10b = 2 12b
) (3 + 5) (9 6) = 3 5 9 + 6 = 12 +
) (2 7) (5 7) = 2 7 5 + 7 = 3
) (11 + 9) (12 + 11 + 9) = 11 + 9 12 11 9 = 12
3) ) ( 15) + (13 + ) = + 15 + 13 + = 28 +
) (3 21) 2 (17 8) = 3 21 2 17 + 8 = 9 38
) (2 4b) (31b 6) 11 = 2 4b 31b + 6 11 = 35b 3
) 14b (15b + y) (y + 10b)=14b 15b y y 10b=11b 2y
5. 1) ) 7 (5 + 8) 11 = 35 + 56 11 = 24 + 56
) 9 + 3 (15 8) = 9 + 45 24 = 15 + 45
) 6 ( + 1) 6 5 = 6 + 6 6 5 = 1
) 19 + 2 (3 4) + 11 = 19 + 6 8 + 11 = 22 + 6
2) ) 13 8 (7 1) = 13 56 + 8 = 43 + 8
) 2 (2 1) + 4 = 4 + 2 + 4 = 4 + 6
) 21 7 4 (9 + 3) = 21 7 36 12 = 15 19
) 33 8 (11b 1) 2b = 33 88b + 8 2b = 90b + 41

79

6. ) ( (3 1)) = + (3 1) = 3 1
) 12 (( ) + 12) = 12 ( ) 12 = 12 + 12 =
= 13 13
) 5 (6 (7 (8 9)) ) = 5 6 + 7 8 + 9 = 2 + 9
) 13b (9b ((c b) 9b)) = 13b 9b + c b 9b = 6b + c
7. ) 0,7b + 0,3b 1,5 = b 1,5; 0,81 1,5 = 2,31
) 1,7 18,7 16,3 = 1,7 35; 1,7 3,8 35 = 28,54
2 25
19
) 2,4 8,4 2 + 0,4 = 0,4 8;
8 =
5 6
3
36
32
68
y 36 64 +
y=
y 100 ;
)
7
7
7
32
102
68 3
102 3500
100 =
100 =
= 102
35
35
35
7 10
8
14
=7
2
16
1
=
) 48 = 16; x =
48
3
1
) 25 = 1; x =
25
3
3
) 2 x = ; x =
7
14
5
5
) x = 2 ; x = 2
8
8
1
1
1
) = 6 x ; x = : ( 6 ) =
6
6
36
1
) x = 3 ; = 9
3
2
3
3 5
3
; x= =
) x =
5
10
10 2
4
) 0,53 = 47,7; = 47,7 : 0,53 = 90

1. ) 2 = 14; x =

2. ) 5 = 0; = 0
) 5 = 10; = 2
) 5 = 15; = 3
5
1
) 5 x = ; x =
9
9
9
9
) 5 x = ; x =
5
25

80

) 5 x =

10
2
; x=
3
3

3. ) 5 = 5 (9) = 45; = 9; ) 7 = 7 0 = 0; = 0
3
3
) x = 15 = 9 ; = 15
5
5

S
S
;b =
b
a
F
F
) a m = F; F = am; a = ; m =
m
a
A
A
) F S = A; A = F S; F = ; S =
S
F

4. ) a b = S; S = ab; a =

1
= 9 ; = 45; 0 = 9; 0 = 9 , 5
, (,
0)
9
2) = 9; x = ; , , 0, c
. = 0, 0 = 9 0 = 9 , .
: = 0
9
9
3) x = ; > 0, , > 0 , , c > 0.
c
c
: > 0.

5. 1) (9) = 9; = 1; c

9
1. 1) ) 3 + 2 = 0;
3 = 2;
2
x=
3
) 3 5 = 0;
5 = 3;
3
x=
5
2) ) 8 5 = 40;
7 = 35;
= 5
) 7t + 21 = t 3;
6t = 24;
t = 4

) 0,6 + 1,8 = 0;
0,6 = 1,8;

= 3
) 7 0,7 = 0;
0,7 = 7;

= 10
) 0,3 5 = 6 0,7
0,3 + 0,7 = 6 + 5
= 11
) 8,31k 71 = 1,11k + 1
7,2k = 72
k = 10
81

) 9 + 13 = 35 + 26
13 = 26; = 2
3) ) 6 + (3 2) = 14
6 + 3 2 = 14
9 = 16
7
x =1
9
) 8 (7 142) = 51
8 7 + 142 = 51
= 91
4) ) (6 + 1) (3 2) = 14
6 + 1 3 + 2 = 14
8 = 16
=2
) (6 2) + 4 = 5 3
6 2 + 4 = 5 3
3 = 13
1
x = 4
3

) 9 + 2,65 = 36,85 9
18 = 34,2; = 1,9
) 5 = 1 (3 9)
5 = 1 3 + 9
9 = 9

=1
) 9 (8 11) = 12
9 8 + 11 = 12
8 = 8
=1
) 12 = (7 9) (11 )
12 = 7 9 11 +
8 = 32
=4
) 11 + 103 = 1 + (12 31)
11 + 103 = 1 + 12 31
= 133

2. 1) 3t + 5 = 5t + 13; 2t = 8; t = 4
6
27
=3
7
7
3 (3t 11) = 5t 17; 9t 33 = 5t 17; 4t = 16; t = 4
1
11 13t = 8t + 11 + 7; 21t = 7; t =
3
0,5t + 3,1 + 8 = 0,5t 4,9
0 = 16 , , t .
81 8,3t (75 8,3t) = 3; 81 8,3t 75 + 8,3t = 3
6 = 3 , , t .

2) 3t + 17 = 2 (5t 5); 3t + 17 = 10t 10; 7t = 27; t =

3)
4)
5)
6)

3. ) (10 3) + (14 4) = 8 (15 22)

10 3 + 14 4 = 8 15 + 22; 2 = 0; = 0
) (2 + 3) (5 + 11) = 7 + (13 2)
2 + 3 5 11 = 7 + 13 2; = 28
) (7 10) (8 8) + (10 + 6) = 8

7 10 8 + 8 + 10 + 6 = 8; 8 = 13; x =

13
5
= 1
8
8

) (2 + 3) + (3 + 4) + (5 + 5) = 12 7
2 + 3 + 3 + 4 + 5 + 5 = 12 7; 17 = 0; = 0

82

3x 2 6 x + 5
=
19
19
1
7
: 3 2 = 6 + 5; 3 = 7; x = = 2
3
3

4. 17 (3 2) = 17 (6 + 5)

5. 8 + 3 = 8 + 7; 9 = 11 ;|| + 3 = 0
10
1.
+ 3
+ + 3 = 67; 2 = 64 = 32
= 32
32 + 3 = 35
2. ; 6
+ 6 = 98; 7 = 98
= 14 ; 6 14 = 84
3. ,
3
+ 3 = 80; = 20 ()
3 20 = 60 () .
4. ; 5
+ 40 = 5; 4 = 40
= 10 () ; 5 10 = 50 () .
5. ; 1,5
2 = 1,5 + 20; 0,5 = 20
= 40 (/)
1,4 40 = 60 (/) .
6. ; 54 ; 54 + 3 = 3 ( + 3); 4 = 48
= 12 ; 54 12 = 42 .
7. ; = 75 + 0,5; = 150 ()
8. ; + 7 ; 36 +
36 + = 3 ( + + 7); 36 + = 6 + 21; 5 = 15
= 3 ; 3 + 7 = 10 ; 36 + 3 = 39 .
11
1.

83

6
5
4

R
O

3
K
-7

-5 -4

P
-3

O
5

C
-3

-4
-5
L1

D
O L

-7

2. (2; 3); (4; 2); (4; 0); D (0; 5); E (4; 3); F (4; 4); H (3; 4);
K (3; 0); L (1; 6); M (0; 1); O (0; 0)
3. 1) (1; 1); (5; 1) (1); (1; 1); D (2; 3) (2)
(1; 2); F (2; 3) (3); G (4; 1); H (2; 2) (4)
2) M (1; 0); M (2; 0) ; N (0; 3); N (0; 2)

D
3
C 2
1

N
A

M
M
-2 -1 0 1 2
E

-1
-3

G
H

-2
F

4. 1) (3,5; 2); (3,5; 2); (1; 4)

D (1; 4); E (3,5; 2); F (3,5; 2); G (1; 4); H (1; 4)
2) : (3,5; 0); M (3,5; 0); : N (0; 4); N (0; 4)
5. 1) : (3,4; 0); : N (0; 2,1) ( = 2,125)

84

A
N

1
-5

-1

2) M (3,5; 0); M (3,5; 0)

6
C

B
A

1
M -3

-4
-6

F
E

6. 1) (25; 360) 1
(2,5; 100) 3
1 1
C ; 4
8 20
2) K (13; b) , b > 0 2
L (a; b) , a > 0, b < 0 4
12
1. 1) = 3 (4) 2 = 14; = 3 (1) 2 = 5
= 3 0 2 = 2; = 3 10 2 = 28

85

8
6 = y = 2 ; = 6; = 6,4; = 9
2
3) = 25; = 0; = 25; = 42,25

2)
2.

0,6 0,3

3
0,3

2
0

1
0,3

0
0,6

1
0,9

2
1,2

= 0; = 0,6; = 0 = 2
3. 1) 1,4 = 28; = 20
2) 5 + 4 = 1,5; 5 = 2,5; = 0,5
4. 1) u = |2,5| 8 = 17; u = |0| 8 = 8; u = |4| 8 = 4
2) u = |3 (25) | = 28; u = |3 0| = 3; u = |3 4| = 1
3) |v| 6u = 3 7u; u = 3 |v|; u = 3 |25| = 22
u = 3 |0| = 3; u = 3 |4| = 1
5. 1) = ; (1; 1); (2; 2); (0; 0)
2) = 0,5; (1; 0,5); (0; 0); (2; 1)
3) = 2 + 1; (0; 1); (1; 3); (11; 23)
6. 1) = 4, = 4
= 0, = 2
= 6, = 10

2) = 4, = 16
= 0, = 0
= 6, = 8

13
1.

1)

0
3

4
1

1
0
-1

= - + 3

2)

86

0
4

2
0


= 2 - 4

1
0

-4

2. 1) )

=+5

= -2 - 6

1
-5

-1 0

-3

-6

= 5 - 2

1
0

-2

2) )
87

1
y = x+2
2

2
1
0

1
0

-4
-6
= 0,3 - 6

0
-1

y=

88

2
x +1
3

3) ) , ) , )

=4

1
0

= -3

-3

= -4,5

-4

3. = 5 2

1)

1
3

1
7

0
2

2) 5 2 = 3; = 1; 5 2 = 1; x =
5 2 = 0; x =
4. 1) (2; 2);
2) (5; 0);
3) (0; 1).

2
8

1
;
5

2
; 5 2 = 2; = 0.
5

=+1
4

y=

1
x +1
3

2
1
(0; 1)
0

89

5. 1) )

y=

x 1
+
2 2

y=

4x
4

1
0

2) )

)
y=

1
(4 x ), x 0
2

= 3( 2), 0

1
0

-6

90

-2

6. 1)

=4

1
-4

=+4

2)

=1

=+1
3

1
-2

1, x 0
7. ) y =
;
3, x > 0

4, x 0
) y =
.
4, x < 0

4
=1

1
0

-3

= -3

-4

91

 14
1.

1)

0
0

2
6

2)

0
0

4
1

= -3

y=
1

1
0

1
x
4

-6

2. 1) )

= 5

= 1,2

1
0

92

1
1

y=

2
x
3

2
1
0

2) )

= -4

y=

1
0

3
x
4

-3
-4

= -0,8

1
0

-4

93

3. = 4
1)

4
16

2,5
1
10
4
1
3
2) 4 = 0; = 0; 4 = 1; x = ; 4 = 3; x = ;
4
4
4 = 5; = 1,25.

4. = 2 y =

1
4

1
x.
2

= -2

y=

1
x
2

1
0

-2

5. (1) = 3; (2) = 3; (3) = 3.

7
1
7
= 2 ; y = x;
3
3
3
9
) y = kx; 9 = 2k; k = = 4,5 ; = 4,5.
2

6. ) y = kx; 7 = 3k; k =

7. 1) )

y=

y=

2,5

x
5
1

0
-1

94

0
-1

y=

2
x
6

1
0

2) )

1
-2

-2

= -

y=x

= 2

-2

-1

y=

1
2

95

 15

1.

2
0
2
1
2
0

0
1
1
1
1
5

2
2
0
3
0
10

4
3
3
3
1
15

2. 1) . 18; (2; 0); N (0; 1);

. 19; (1,5; 0); N (0; 3);
. 20; (2; 0); N (0; 10);
2) . 18; = 1; = 0; = 5;
. 19; = 2; = 4; = 12;
. 20; = 0; = 2; = 5.
3. 1)
2)
3)
4)

2
3
1
5
6
20

6
4
3,5
4
5
15

6
2
0,5
1,5
2
0

8
3
2,5
2
2
20

(1)
(2)
(3)

3) . 18; = 3; = 5; = 7;
. 19; = 1; = 0; = 5;
. 20; = 3; = 5; = 8;

15 ;
0,5 5 12,5 ; 1 10 10 ;
1 ; 10 ;
1,5 .

15
15
2) V =
= 10 /;
= 5 /;
1,5
3
3) : = 10. : = 15 5.

4. 1) V =

16

1
;
3
2) ;
3) 1 (3; 0); N1 (0; 1); M2 (5; 0); N2 (0; 2); M3 (0; 0); N3 (0; 0).

1. 1) k1 = k 2 = k 3 =

y=

y=

1
3

x+2

1
0
-1

y=

96

1
3

x 1

2.

= -2 4

=4
1
-2

= -4

-4

1) 1 (4; 0); N1 (0; 4); M2 (2; 0); N2 (0; 4);

M3 ; N3 (0; 4);
Mi c OX; Ni c OY;
2) (0; 4).
3.

= 2 + 4

= -2 + 4

= -2 4

= 2 4
4

1
-2

-4

M1 (2; 0); N1 (0; 4);

M2 (2; 0); N2 (0;4);
= 2 + 4 = 2 4;
= 2 + 4 = 2 4.

M3 (2;0); N3 (0; 4);

M4 (2; 0); N4 (0; 4);

97

y = 6x 3 ;
4.

y = 3 x + 6
6 3 = 3 + 6;
9 = 9;
= 1;
= 6 1 3 = 3;
(1; 3)
y = 5 x 2 y = 5x + 2

= 6 3

3
1
0

1 2

-3
= -3 + 6

2
2
x + b ; 4 = 4 + b; b = 0; y = x ;
5
5
) = 3x + b; 8 = 9 + b; b = 17; y = 3x + 17.

5. ) y =

6. = 0,5x + b;
2 = 2 + b; b = 4;
y = 0,5x 4;
y = 0,5x + b ;
1 = b;
= 0,5 + 1.

1
0

-1
-3
-4
= 0,5 - 4

7. . 22;
= 3 (0; 0);
. 23;

1
5

y = x , .. k =

1
< 0;
5

. 24;
= 2 + 3 (0; 3) ,
.
98

 17
1. 1) ) 10 ;
= 35 5 5 = 10 , 15 , ;
) 25 ;
) 17,5 ;
) 35 ;
2) , ; = kx + b, y = d; x = m; k = 5; b = 35;
3)
d

35
30

d = 35 5m

25
20
15
10
5
0

4) ) 26,5 ; ) 22,5 ; ) 17 ; ) 35 ;
5) ) 3 ; ) 5,8 ; ) 4,2 ; ) m = 0 .
2. 1) ) 10,2; ) 16; ) 11; ) 11,7;
2) , , k = 0,01; b = 10;
3)
t
l
t = 10 +
100
18
16
14
12
10
8
6
4
2
0

100 200

800

4) ) 11,5; ) 12,2; ) 14,6; ) 10;

5) ) 0 ; ) 500 ; ) 700 ; ) 100 ;
6) ) 2; 1; ) 2,5.
99

 18
3

17
1
125
5
1. 1) ) 64; ) 0,49; )
; ) =
=4
;
27
27
81
3
5

1
1
1
;
2) ) 36; ) 0,0081; ) 1; ) = 5 =
32
2
2
33
27
3) ) 81; ) 3 =
; ) 0,24 = 0,0016.
125
5
2. 1) (8,6) 3 = 8,63 < 0;
2) (1,24) 2 = 1,242 > 0;

3) 362 < 0;
4) 0,453 = 1 0,453 < 0.

3. 1) 31; 33; 34; 35;

3)

2) (0,1) 5; (0,1) 3; 0,11;

4. 1) ) 0,2 4900 = 980 ;

) 0,0016 : 40 = 0,00004;
2) ) (2,5) 2 = 6,25;

1 1
1 1
= ;
= ;
9 3 81 3
4) (2) 1; (2) 3; (2) 4.
2 4
8
= ;
3 9
9
) 0,5 16 = 8;
) 64 + 8 = 72;
) 3

) 0,25 = 0,00032;
4

) 81 36 = 45; ) 125 1000 = 1125; ) 1 + 1 = 0.

) 3,048625; ) 33,1776; ) 3,15 = 286,29151;
) 9,16; ) 238,328; ) 50,002.
0,42 + (0,5) 2 = 0,16 + 0,25 = 0,41; 3) 1,22 0,82 = 0,8;
(4,8 + 3,9) 2 = 0,81;
4) (2,6 1,8) 2 = 0,64.
) 12 = 9 3 = 729 ;

3)
5. 1)
2)
6. 1)
2)

3 3 5
+ 0,36 = 0,6 + 0,36 = 0,96 ; ) 2000 0,0081 16 = 0,2;
5 5 3
1,8
9
2) )
+ 64 = 84 ; ) 8 + 1 ( 27 ) = 4,5 27 = 22,5 .
0,09
16

7. 1) )

8. 1) ) (4,8) 4 (5,7) = 5,7 4,84 < 0;

) (9,4) 5 : (3,1) = 9,45 : (3,1) > 0;
2) ) (4,5) 3 (3,8) 2 = 4,53 3,8 > 0;
) (8,4) 6 : (2,3) 4 = 8,46 : 2,34 < 0.
9. 1) ) (6,1) 5 < (2,3) 4, .. 6,15 < 2,34;
) (1,3) 5 (2,4) > (3,8) 5 : (0,7) 2, .. 1,35 2,4 > 3,85 : 0,72;
2) ) (0,4) 6 > (0,4) 8, .. 0,46 > 0,48;
) (2,3) 5 > (2,3) 7, .. 2,35 > 2,37, .. 2,35 < 2,37.

100

 19

25 49
4
;
=5 ;
36 9
9
8
;
64; 125; 0,064;
27
7,2; 0,032; 0,45;
270; 0,27; 2160;
67; 14; 107;
0,096; 1100.

1. 1) 81; 49; 0,36;

2)
3)
4)
5)
6)
2.

1)

2
2
2
+ 3,5

2)

3
0,53
3 10

3. 1)

8
64
64
67,5
6
216
108
226

1
1
1
4,5
0,2
0,008
0,004
10,008

0
0
0
3,5
0
0
0
10

0,9
0,81
0,81
4,31

1,5
2,25
2,25
5,75

1
1
0,5
9

14
196
196
199,5

8
512
256
502

12
3
= ; 10 (1) 3 = 10;
64
16
2

2)
3)
4)
4. 1)

2)

9
0;
= 36 ;
1,5
(1,6 + 2,1) 3 = 0,125; (1,4 + 2,6) 3 = 64;
(9,5 10,3) 2 = 0,64; (0,4 + 0,6) 2 = 1.
(3) 2 < 32, .. 32 < 32;
02 = (0) 2, .. 0 = 0;
42 < (4) 2, .. 42 < 42;
() 3 = 3;
, = 4; 0; 5.

5. 1) ) 2 0;
) ( 4) 2 0;
) 2 0;
) 2 + 1 > 0;
) 2 5 < 0;
2) ) 2 + b2 0;
) a2 + b2 + 4 > 0;
) (a + b) 2 0;
) (a + b) 2 0.

101

 20
1. 1) ) 8 3 = 11;
) 4 4 = 8;
) 2 = 3;
) 57 54 = 511;

2) ) 3 2 = 6;
) 9 2 4 = 15;
) (4) 3 (4) (4) 6 = (4) 10 = 410.

2. 1) ) 10 : 5 = 5;
) b7 : b6 = b1 = b;
) 8 : 7 = 1 = ;
) 9 : 9 = 0 = 1;

2) ) 821 : 89 = 812;
) (0,3) 12 : (0,3) 5 = (0,3) 7;
) (0,2) 16 : (0,2) 6 = (0,2) 10 = 0,210.

3. 1) 4 8 = 12; 2) 3 = 4; 3) 14 : 7 = 7; 4) 19 : 9 = 10.
4. 1) 8 3 : 5 = 8+35 = 6;
2) 20 : 10 = 2010+1 = 11;

3) 7 : 3 : 3 = 733 = 1 = ;
4) 14 : 9 5 = 149+5 = 10.

5. 1) 816+518 = 83 = 512;
2) 101015 = 104 = 10000;

3) (2) 7+48 = (2) 3 = 8;

4) 0,310+786 = 0,33 = 0,027.

6. 1) 2 an = an+2;
2) x xm = xm+1;
3) y12 : yn = y12n;

4) cm : c3 = cm3;
5) a2n an = a2n+n = a3n;
6) x2n : xn = x2nn = xn.

8. 1) ) 22 (18 : 9) = 22+ (189) = 31;

) 16 (12 4) = 16+ (12+4) = 32;
) 18 : (18 : 9) = 18 (189) = 9;
2) ) (8 2) : (4 5) = (8+2) (4+5) = 1 = ;
) (25 : 5) 10 : 3 = (255) +103 = 27.
9. 1) (12) 6 (12) 5 = (12) 11 = 1211 > 0;
2) (4) 16 : 46 = 416 : 46 = 410 > 0.

am
= a m2 ;
a2
2) a2n a2n = a4n; a5n : an = a4n;
3) an1 a = an; a2n : an = an.

10. 1) m4 a2 = am2;

21
1. 1) ) (bc) 6 = b6c6;

) (3) 3 = 2733;
4

) (abc)

10

10 10 10

=a b c ;

) (2) 5 = 255 = 325;

2) ) (4) 3 = 643;

) xyz = 0,0001x 4 y 4 z 4 ;
10

) (5) 2 = 2522;
3

) (0,1) 4 = 0,00014 ;

102

27
3

abc = a 3 b 3 c 3 .
64
4

2. 1) ) () 2 = (1 ) 2 = (1) 2 2 = 2;
) () 6 = (1 ) 6 = (1) 6 6 = 6;
) (200) = (1 ) 200 = (1) 200200 = 200;
) () 2n = (1 a) 2n = (1) 2na2n = a2n;
2) ) () 3 = (1 ) 3 = (1) 33 = 3;
) () 5 = (1 ) 5 = (1) 55 = 5;
) () 23 = (1 ) 23 = (1) 2323 = 23;
) () 2n+1 = (1 a) 2n+1 = (1) 2n+1a2n+1 = a2n+1.
3. 1) ) a6b6 = (ab) 6; ) 4922 = (7) 2; ) 0,00014b4 = (0,1ab) 4;
2) ) 3 = () 3; ) 273 = (3) 3; ) 3255 = (2) 5;
3) ) 7y7z7 = (xyz) 7; ) 0,0016a4c4d4 = (0,2acd) 4;
3

1 3 3 3 1

a b c = abc .
8
2

4. 1) 43 53 = (20) 3 = 8000;
4

1
1

2) 30 4 = 30 = 625 ;
6
6

3) 803 0,53 = (80 0,5) 3 = 403 = 64000;

1
3

10 3
= 5 4 = 625 .
3
2

4) 3 1,5 4 =
5. 1) ) (2) 6 = 12;
) (3) 3 = 9;
) (5) 4 = 20;
) (xn) 3 = x3n.

2) ) (5) 2 = 10;
) (4) 3 = 12;
) (a3) 2n = a6n;

6. 1) (4) 4 = 16; 2) (6) 2 = 12; 3) (2) n = c2n ; 4) (cn) 3 = c3n;

7. 1) ((3) 4) 5 = 345 = 60;
2) ((2) 2) 2 = 222 = 8;
3) ((3) 3) 3 = 333 = 27.

4) (() 2) 3 = 23 = 6;
5) ( () 3) 2 = (3) 2 = 32 = 6;

8. 1) ) (33) 4 = 312; ) ((32) 3) 2 = 312;

2) ) ((3) 2) 2 = (3) 4; ) ((9) 2) 3 = 96 = (32) 6 = 312.
9. 1) () 2 = (2) = 2; 2) () 3 = (3) = 3.
22

) 2 () 5 = 7;
1. 1) ) () 2 5 = 2+5 = 7 ;
) 2 5 = 7;
) (2) (5) = 7;
2) ) (3) 2 4 = 10;
) 3 (3) 3 = 3+33 = 12;
3
5 4
(5+3) 4
32
) ( ) =
= ; ) ( 5) 5 = (1+5) 5 = 30;
3 2
2 3
32+23
= 12;
3) ) ( ) ( ) =
) (3 ) 3 (3 ) 2 = (3+1) 3+ (3+1) 2 = 20;
) (6) 2 (4 2) 2 = 62+ (4+2) 2 = 24;
103

4) ) 10 : (2) 5 = 1025 = 0 = 1; ) (3) 7 : (3) 6 = 3736 = 3;

) (2 ) 3 : (3 ) 2 = (2+1) 3 (3+1) 2 = 1 = .
2. ) 2 (4) 2 = 10; ) (6) 6 : 2 : (17) 2 15; ) () 2 () 3 : 2.
3. 1) ) 28 (23) 2 : 212 = 28+3212 = 22 = 4;
) 715 : (75) 2 : 73 = 715523 = 72 = 49;
2) ) 162 : 25 = (24) 2 : 25 = 285 = 23 = 8;
) (33) 4 : (32) 5 = 31210 = 32 = 9;
) 323 82 : 165 = (25) 3 (23) 2 : (24) 5 = 215+620 = 21 = 2;
3) ) 310 710 : 218 = (3 7) 10 : 218 = 21108 = 212 = 441;
) 615 : 213 : 313 = 615 : (2 3) 13 = 61513 = 62 = 36;
) 2010 : (510 410) = 2010 : (5 4) 10 = 201010 = 200 = 1.
4. 1) (3 ) 3 : 6 = 6;
2) (4) 3 3 = 15;

3) (4) 3 () 3 = 15;
4) (3 2) 2 = () 7 () 3.

5. ,
, ,
, , ,
00 .
23
1.

1)

1,52

2
6

0,8
0,96

0
0

1
1,5

20
600

2)

53

10
5000

0,4
0,32

0
0

2
40

8
2560

4
= 7 ;
3
4) 0,04 15 (2) 2 = 2,4; 0,04 (8) (10) 2 = 32;
5) 0,1 (1) 1 (20) = 2; 0,1 3 (4) (2) = 2,4.
3) 3 (2,5) 8 = 60; 3 1,75

2.

1)

2)

82

0,5
2

0,4
1,28

0,3
0,72

82

0,3
0,72

0,4
1,28

0,5
2

10
8
6
0,53 500 256 108

0,53

104

6
108

8
256

10
500

0,2
0,32

0,1
0,08

0
0

0,1
0,08

0,2
0,32

4
32

2
4

0
0

2
4

4
32

3. 1)
2)
3)
4)

1,7 2,1 0,8 5,6 = 15,9936;

0,8 1,42 2,53 = 24,5;
8,5 11,52 12,42 (8) = 1382763,68;
3,7 (1,8) 3 4,53 8,1 = 15927,28677.

a =1
2)
b = 6 ; 5ab = 30;

4. 1) 0,3 = 0; = 0;

a = 0,5 ; 5ab = 10;

b = 4

0,3 = 0,6; = 2;
0,3 = 0,8; a = 2
0,3 = 1; a = 3
5. 1)
2)
3)
4)

2
;
3

a = 0 ; 5ab = 0;
b = 11

a = ; 5ab = 5.

7
b = 7

1
;
3

, 2 (1) 3 = 2 <0;
, 10 06 = 0 ;
, 0,032 0, .. 2 0;
, 2,72 0, .. 2 0.

24

3
y = 6 x 2 y ; ) b3 3b2 = 3b5;
4

1. 1) ) 12 0,5 = 62; ) 8 x 2

3 2
xy 16 y = 12 xy 3 ; ) 1,62 (22) = 3,233;
4
) 34 1,465 = 1,499.
2. 1) 204 0,52 (0,323) = 375;
3

2) 12 x 2 y 2 z xy 2 z 2 0,1x 2 yz 2 = 0,9 x 5 y 5 z 5 .
4

2) )

3. 1) 7,5 42 = 303; 2) 8a2b4 (a3b2) = 8a5b6.

3

1
1 2
a = a 6 ; ) (0,15) 4 = 0,000120;
8
2
2) ) (5) 3 = 12533; ) (44) 3 = 64312; ) (553) 3 = 125159;

4. 1) ) (6) 2 = 362; )
4

1 4 4
1
xy =
x y ; ) (1026) 3 = 1000618;
81
3
) (2b3c4) 7 = a14b21c28;
4) ) (3a2b) 3 = 27a6b3;
) (2ab4) 3 = 8a3b12;
) (3b2c) 4 = a12b8c4.

3) )

105

1 6 1 3
4 10
2 5 2
a = a ; 0,16 b = (0,4a b ) ;
9
3
2) 0,008x9 = (0,2x3) 3; 27a3b12 = (3ab4) 3.

5. 1)

6. 1) ) 35 (2) 2 = 35 42 = 1403;
) 43 (52) 3 = 43 1256 = 5009;
) (42) 3 5 = 646 5 = 6411;
4
1
1

2) ) x 2 y 3 2 x 6 y = x 2 y 3 16 x 24 y 4 = 2 x 26 y 7 ;
8
8

100 2 12
1

) 90a 4 b 3 3 ab 6 = 90a 4 b 3
a b = 1000a 6 b15 .
3
9

7. 1) ) (102) 2 (32) 3 = 10042 2736 = 270078;

3

( )

2
1
1

) xy 3 4 y 5 = x 3 y 9 16 y 10 = 2 x 3 y 19 ;
8
2

2) ) (362) 3 (2) 4 = 27186 84 = 272610;

) (5ab6) 4 (0,2a6b) 4 = 625a4b24 0,0016a24b4 = a28 b28;
8. 1) ) , , (7a3b2) 2 = 49a6b4;
) , .. ,
2524 0, .. 24 = (2) 2 0;
2) ) 0,1a4b2 (10a2b4) = a6b6; a6b6 = (a3b3) 2, ;
) (24) 3 2b8 = 8a12 2b8 = 16a12b8 = (4a6b4) 2, .

C 25
1. 1) ) b ab + a2b = ab2 + a2b;
) 5 82 72 3b = 40xy2 21x2b;
) 3 8b c2 c 2a = 24bc3 2ac;
) 5 8 (72) + (6) 32 = 2803 182;
2) ) 2m6 + 5m6 8m6 11m6 = 12m6;
) 3,12 + 2,12 2 = 22;
) 12ab 5ab 2ba = 5ab;
) 96 + 2 132 92 = 122.
2. 1) ) 8b3 3b3 + 17b 3b3 8b 5 = 2b3 + 9b 5; n = 3 ();
) 52 + 3 7 53 32 + 7 11 = 53 + 22 + 10 18; n = 3;
) 4 3 + 2 + 1 + 3 2 + 1 = 4; n = 4;
2) ) 32 + 5 72 + !22 6 = 152 72; n = 2;
) 92 8 62 92 = 62 9; n = 2;
) 6a2b 5ab2 + 5a3 + 2ab2 8a3 3a2b = 3a3 + 3a2b 3ab2; n = 3.
3. 1) 15 b 2 + 14a = a b 2; 29 + 2 2 = 29;
2) 6 + + 7 = 5 + 7; (3) (3) + 15 21 = 3;
3) m4 3m2n + m2n2 m3n 4mn3 = m4 4m3n + m2n2 4mn3;
1 + 4 + 1 + 4 = 10.

106

4. 1) 32 52 + 2 15;
2) 33 52 2 15;

3) 243 202 + 4 15;

4) 246 204 + 42 15.

5. ) 82 7 52 42 20 52 + 22 + 7 + 32 = 2 20;
) 32a3bc 23ab3c 37abc3 35ab3c + 36abc3 33a3bc =
= a3bc 58ab3c abc3.
6. ) 3 22 + 3 1 + (6);
) 27 5 + 24 + (6 27);

) 6 24 3 + (1);
) 33 23 + 6 + (1 6).

7. ) 8b + 13 5b 37 11b + 35 + 8b = 11;
) 8b2x2 5x3 + 3x 17x2b2 + 5 10x + 9x2b2 = 5x3 7 + 5;
) 23 5by + b2 + 7y2 + 3by 5b2 + 9y2 + 2by + 4b2 = 18y2.
8. : 1) 6 + 4 + 2 + 3; 2) 2 + b2 + 2;
: 1) 32 1; 2) 2 b2 a2b2 16.
26
1. 1) ) 32 + 7 5 + 32 + 1 = 62 + 7 4;
32 + 7 5 32 1 = 7 6;
) 5 + 3 22 + + 7 = 22 + 6 + 10;
5 + 3 + 22 7 = 22 + 4 4;
) + 6 + 3 6 = + 3;
+ 6 3 + 6 = 12 + 3;
) 2 3 + 2 + 2 2 = 22 3;
2 3 + 2 2 + 2 = 22 3;
2) ) 52 3 1 + 82 + 2 11 = 132 12;
52 3 1 82 2 + 11 = 32 5 + 10;
) 22 + 3 2 + 53 3 + 2 = 53 + 22;
22 + 3 2 53 + 3 2 = 53 + 22 + 6 4;
) 3 3 + 15 + 3 + 3 15 = 23;
3 3 + 15 3 3 + 15 = 6 + 30;
) 82 + 2 32 + 22 + 3 32 = 102 + 5 62;
82 + 2 32 22 3 + 32 = 62 .
2. ) (3 + 5b) + (9a 7b) + (5a + 11b) = 3a + 5b + 9a 7b 5a + 11b =
= 7a + 9b;
) (2 11) (5 + 12) + (3 17) = 2 11 5 12 + 3
17 = 40;
) (3b2 + 2b) + (2b2 3b 4) (b2 + 19) = 3b2 2b + 2b2 3b 4 +
+ b2 19 = 6b2 b 23;
) (a b + c) + (a c) (a b c) = a b + c + a c a + b + c = a + c.
3. 1)
2)
3)
4)
5)

1 ();
1 1 = 2 ();
+ 1 = 2 1 ();
2 + 3 = 2 5 ();
+ 1 + 2 + 3 = 4 6.

107

4. 1) 35 33 + 8 35 + 33 + 8 = 0;
35 33 + 8 + 35 33 + 8 = 65 63 + 2 16;
2) 27b3 27b2y + 9by2 y3 + 20b3 + 27b2y + 9by2 3y3 =
= 47b3 + 18by2 4y3;
27b3 27b2y + 9by2 y3 20b3 27b2y 9by2 + 3y3 =
= 7b3 54b2y + 2y3.
5. 1) (2x + 3y 5z) (6x 8y 3z) + (5x 8y 9x) = 2x + 3y 5z 6x +
+ 8y + 3z + 5x 8y 9z = x + 3y 11z;
2) (2k3 k2 k + 1) (6k4 3k3 3k2 + 21k) (2k5 k4 k3 + 2k2) =
= 2k3 k2 k + 1 6k4 + 3k3 + 3k2 21k 2k5 + k4 + k3 2k2 = 2k5
5k4 + 6k3 22k + 1.
6.

1
1)
5 + 1
2)
22 + + 3
3)
3 32b 5b3
4)
2 + 5 3
5)
2 2 2
6)
2 + 3
1 + = 2; = 2 1.

2
4 4
22 +
0
2 5 + 3
4 + 22
2 2

3
9 3
2 + 3
a3 3a2b 5b3
0
2
+ 2 + 2
2 + 2

27
1. 1) ) 2bx + 2by + 2x + 2y = (2bx + 2by) + (2x + 2y);
) b3 b2 b + 3y 1 = (b3 b2 b 1) + (3y);
2) ) bx2 x + 1 b = (bx2 b) + (1 x);
) a2 b2 2ab 1 = (b2 2ab) + (a2 1).
2. 1) ) ab c + b = (ac ab) (c b);
) am + an + m n = (am + an) (n m);
) 2 + 5 5 = (2 5) (5 );
2) ) 2 1 + + = (2 ) (1 );
) + + 3 = ( ) ( + 3);
) 2b + a2 b2 1 = (a2 1) (b2 2b).
3. ) + 1 = ( + ) ( + 1);
) 8 3 1 + 24 = (8 + 24) (3 + 1);
) 3z 5y 2 = 3z (5y + 2);
) 3a 5b + 8 = 8 (3a + 5b).
4. ) (52 3 2) (8 8 2) + (3 + 7) = 52 + 4 5 + 8 =
= (52 + 4 5) + 8;
) (32 8by c) (5a2y + 4by 3c) 5c = 3a2y 8by c 5a2y
4by + 3c 5c = 2a2y 12by 3c = (1 3c) (2a2y + 12by + 1).

108

C 28

) a (k + c 3) = ak + ac 3a;
1. 1) ) p (a + b) = pa + pb;
) y (k + c) = yk yc;
) ( b + 1) = xa + xb x;
2) ) 52 (2 ) = 102 53;
) 73 (5 + 3) = 78 214;
3
2
4
5
) 8b (b 2b ) = 8b + 16b ; ) (15 + 20) 1223 = 1238 + 1243;
3) ) 2m4 (m5 m3 1) = 2m9 2m7 2m4;
) 3 (3 + 4) = 34 32 + 12;
) (82 4 + 16) 0,25 = 23 2 + 4;
) 2 (32 + 5 2) = 63 + 102 22;
) b5 (b6 5b3 + b 3) = b11 5b8 + b6 3b5;
) 9 (24 + 2 2 + 1) = 185 93 + 182 9.
2. 1) ) (a + b) p = ap + bp; ) k (m n) = km + kn;
2) ) ( + ) = + ; ) (x + y + z) (bc) =
= bcx bcy bcz;
3) ) 2 (2 ) = 22 3;
) ( 1) 2 = 22 2.
3. 1) ) 5 ( + 2) + ( + 2) = 5 + 10 + + 2 = 6 + 12;
) ( 3) 3 ( 3) = 3 3 + 9 = 2 + 6;
) 7 ( 7) 3 ( 3) = 7 49 3 + 9 = 4 40;
) 15 (8 1) 8 (15 + 4) = 120 15 120 32 = 47;
2) ) 2 ( + 1) 4 (2 ) = 22 + 2 8 + 42 = 62 6;
) 2 (2 3) 3 (5 3) = 4 62 152 + 9 = 212 + 13;
) 3 (c + d) + 3d (c d) = 3c2 + 3cd + 3cd 3d2 = 3c2 + 6cd 3d2;
) 5b (3a b) 3a (5b + a) = 15ab 5b2 15ab 3a2 = 5b2 3a2;
3) ) (2 + ) (2 + + 1) = 3 + 2 2 1 = 3 1;
) 22 (6 1) + 3 ( 42) = 123 22 + 32 123 = 2;
) (22 3n) n (2n2 + a) = 2a3 3an 2n3 an = 2a3 4an 2n3;
) b (b3 b2 + b) (b3 b2 + b) = b4 b3 + b2 b3 + b2 b =
= b4 2b3 + 2b2 b.
4. 1) (2 2) + (3 ) 2 ( 2) = 2 22 + 3 2 2 + 22=
= 5 2 2;
5 (0,1) 0,7 (0,1) 2 + 0,2 = 0,16;
2) 2 (2 + 5 1) 3 (3 + 52 ) + 24 + 103 22 = 4 + 53 2
34 153 + 32 + 24 + 103 22 = 0,

1
3

, 0 p = 3 .
5. 1) ) (4 3b + a2b2 ab3) a2b = a6b a5b2 + a4b3 a3b4;
) 2k2x3 (3x3 + 2k2 k k2) = 6x6k2 + 4x5k2 2k3x3 2k4x3;
2) ) 5 (33 2 + 3) = (154 53 52 + 53) =
= 154 53 522 + 54;
) ab (a2b ab2 a3b3) p = a3b2p + a2b3p + a4b4p.

109

C 29
1. 1) ) (2 7) + (6 + 1) = 18;
2 7 + 6 + 1 = 18;
8 = 24;

= 3;

) 24 2 (5 + 4) = 6;
24 10 8 = 6;
10 = 10;
= 1;
2) ) 20 + 4 (2 5) = 14 + 12;
20 + 8 20 = 14 + 12;
6 = 12;
= 2;
) 15 1 = 3 (7 1) 2;
15 1 = 21 3 2;
6 = 4;
4 2
x= = ;
6 3
3) ) 5 (2 7) = 0;
10 + 35 = 0;
35 = 10;
2
x= ;
7
) 5 (2 7) = 5;
10 + 35 = 5;
35 = 15;
3
x= ;
7
2. 1) 3 11 = 0;
3 = 11;
2
x = 3 ;
3
2) 0,3 10 = 4 0,7;
= 14;

) (4 8,2) (3,8 + 1) = 5;
4 8,2 3,8 1 = 5;
12 = 2;
2
1
x= = ;
12
6
) 12 = 6 + 6 (3 1,5);
12 = 6 + 18 9;
18 = 27;
= 1,5;
) 8 (11 2) + 40 = 3 (5 4);
88 + 16 + 40 = 15 12;
= 36;
) 2 12 (3 ) = 1 + 3 ( + 2);
2 36 + 12 = 1 + 3 + 6;
11 = 43;
10
x=3 ;
11
) 8 (5 1) = 0;
40 8 = 0;
40 = 8;
1
x= ;
5
) 8 (5 1) = 8;
40 8 = 8;
40 = 16;
2
x= .
5
3) 3 6 + 16 = 2 + 3;
8 = 16;
= 2;

4) 3 ( + 1) = 5 + 12;
3 + 3 = 5 + 12;
2 = 9;
= 4,5.

3. 1) ) 4 (2 3) + 7 (6 + 1) 9 (9 + 4) = 30;
8 12 + 42 + 7 81 36 = 30;
51 = 51; = 1;

110

) 17 2 ( + 3) + 5 ( 7) 3 (2 + 1) = 28;
17 2 6 + 5 35 6 3 = 28;
1
3 = 1; x = ;
3
) (4 + 11) 7 (2 5) = 3 ( + 3);
42 + 11 72 + 35 = 32 9;
55 = 0; = 0;
2) ) n (12 n) 5 = 4n n (10 + (n 3));
12n n2 5 = 4n 10n n2 + 3n;
1
15n = 5; n = ;
3
) 16 + 5 ( 2 ( 4)) = 12 (3 2) 1;
16 5 10 + 40 = 36 24 1;
2
9 = 11; c = 1 .
9
4. 1) 2 + 11 = 1 ();
2) 1 = 2 (); 1 (4) = 3 = 2 (4);
1 (10) = 31; 2 (10) = 11; 1 (10) 2 (10) , .
30
1 4x
= 1 ; 1 4 = 5; 4 = 4; = 1;
5
3 x 10
2
= 1 ; 3 10 = 2; 3 = 8; x = 2 ;
)
2
3
x+3 1
)
= ; + 3 = 2; = 1;
10
5
8 x + 3 10 x 1
; 8 + 3 = 10 1; 2 = 4; = 2;
2) )
=
7
7
x + 2 3x 5
=
; 4 ( + 2) = 5 (3 5); 4 + 8 = 15 25;
)
5
4
11 = 33; = 3;
7 x 19 x 11
=
; 4 (7 ) = 3 (19 11);
)
6
8
28 4 = 57 33; 61 = 61; = 1;
5x 9 5x 7
2x + 3 x 6
=
+
= 1;
) 2 x
;
3) )
4
4
3
3
5 9 + 5 7 = 4;
6 2 3 = 6;
10 = 20;
3 = 3;
= 2;
= 1;

1. 1) )

111

2 x x 1

= ;
5
15 3
3 (2 ) = 5;
6 3 = 5;
4 = 1; = 0,25;
2x 3 x 1
+
=2;
)
9
5
5 (2 3) + 9 ( 1) = 90;
10 15 + 9 9 = 90;
19 = 114;
)

= 6;

x 3x 1

=2;
7
14
2 3 + 1 = 28;
= 27;

x + 14 6 x + 1

= 1;
5
7
7 ( + 14) 5 (6 + 1) = 35;
7 + 98 30 5 = 35;
23 = 58;
12
x=2
.
23

5 x 4 3x 2 2 x 1
+
+
= 3x 2 ;
3
6
2
2 (5 4) + 3 2 + 3 (2 1) = 6 (3 2);
10 8 + 3 2 + 6 3 = 18 12; = 13 12 = 1;
2x 3 x 1 5x + 1
2)
+
+
= 3 x ;
5
4
20
4 (2 3) + 5 ( 1) + 5 + 1 = 20 (3 );
8 12 + 5 5 + 5 + 1 = 60 20; 38 = 76; = 2;
3x 2 5 x 7 1
3) x 2 5 x + 3
= ;
3
3
3 (2 5 + 3) 32 + 5 + 7 = 1; 32 15 + 9 32 + 5 + 7 = 1;
10 = 15; = 1,5.

2. 1)

C 31
1. 1) 5 + 11 + 3 5 = 17; 8 = 11;
2) 5 + 11 = 3 5 + 13; 2 = 3;
3) 5 + 11 = 3 5 + 13; 2 = 3.

4) 2 (5 + 11) = 3 5; 7 = 27;
5) (5 + 11)=3 5 + 13; 12=25;

2. 1) ; + 30 ;
3 ( + 30) + 2 = 240;
3 + 90 + 2 = 240;
5 = 150;
= 30 / ;
30 + 30 = 60 (/) ;
2) 3 ; 8 5 ;
3 = 5 ( 8);
3 = 5 40;
2 = 40;
= 20 3 ;
3 20 = 60 () ;

112

3) - ;
+ 30 ;
20 ;
5 ( + 30) + 4 + 3 ( 20) = 6090; 5 + 150 + 4 + 3 60 = 6090;
12 = 6000;
= 500 () ;
500 + 30 = 530 () ;
500 20 = 480 () ;
4) ; + 16 ;
4 ( + 16) + 3,5 = 94; 4 + 64 + 3,5 = 94;
7,5 = 30;
= 4 (/) ;
4 + 16 = 20 (/) ;
( 4 . 4 , 0,5 4 0,5 = 3,5 ).
5) .
;
1 : + 6 ;
+ 2 ( + 6) = 39; 3 = 27; = 9 ();
9 + 6 = 15 () ;
2 : 6 ;
+ 2 ( 6) = 39; 3 = 51; = 17 ();
17 6 = 11 () .
32

) n (1 3m) = n 3mn;
1. 1) ) (3 + 2) = 3 + 2;
) b (2a 5) = 2ab 5b;
) ( + 1) = ;
2) ) 7 (b 2a) = 7ab 14a2;
) 25 ( 2) = 25 502;
) 5 ( + 3) = 52 + 15;
) 6b (5y2 + b) = 30by2 6b2;
3) ) 4 ( 1) = 5 4
) 5 (1 + 3 + 42) = 5 + 36 + 47;
3
3
6
3
) 2m (m + 4) = 2m + 8m ; ) 32 (1 2 + 62)=32 63 + 184;
4) ) bc (5c + 1) = 5bc2 + bc;
) ab (ab 4b2 + 6a2) = a2b2 4ab3 + 6a3b;
) 422 (22 3) = 842 1222;
) 322 ( + 2 3) = 332 + 623 933.
2. 1) ) ( + ) + (b x) = a (x + y + b x) = a (b + y);
) b (2x 5y) b (3x y) = b (2x 5y 3x + y) = b (x 4y) =
= b ( + 4);
) 2 (a + b) + c (5a 3b) = c (2a + 2b + 5a 3b) = c (7a b);
) 2 (2 + 7) 2 (3 5) = 2 (2 + 7 3 + 5) = 2 (12 );
2) ) a (b + c) + x (b + c) = (b + c) (a + x);
) a (3b + c) x (3b + c) = (3b + c) (a x);
) 3 (2 9) 5 (2 9) = (2 9) (3 5);
) 2 (3 + 1) + (3 + 1) = (3 + 1) (2 + 1);

113

3) ) k (x y) + c (y x) = (x y) (k + c);
) 3 ( ) ( ) = ( ) (3 + 1);
) 2 ( ) ( ) = ( ) (2 + ) = 3 ( );
) ( ) + b (a y) = (y a) (1 b).
3. . 27. , 2r
r
r 2 r 2
S = 2ra

= 2ra r 2 = r (2a r ) ;
2
2
. 27.
2r r. S = 4r2 r2 = r2 (4 ).
4. 1) ) 352 + 1543 + 1234 = 332 (2 + 5 + 42);
) 7a3b3 77a2b3 21a3b4 = 7a2b3 (a 11 3ab);
) 5322 1532 54 = 53 (2 3 );
2) ) ( + 5) (2 + 1) + ( + 5) (3 8) = ( + 5) (2 + 1 + 3 8) =
= ( + 5) (5 7);
) (5m 3) (n + 1) (2n + 3) (3 5m) = (5m 3) (n + 1 + 2n + 3) =
= (5m 3) (3n + 4);
) (2a b) (3a + 11) + (5a 11) (b 2a) = (2a b) (3a + 11 5a +
+ 11) = (2a b) (22 2a) = 2 (b 2a) (a 11).
5. 2 5 1 = 7; 2 5 = 8;
1) 3 (2 5 1) = 3 7 = 21; 2) (2 5 1) (2 5) = 7 8 = 56;
3) 9 (2 5) 7 = 9 8 7 = 65.
33
1. 1) ) ( + 4) ( 5) = 5 + 4 20;
) ( 8) (6 ) = 6 48 + 8;
) (10 ) ( + 3) = 10 30 3;
) (2 ) ( 9) = 2 + 18 + 9;
2) ) ( + 3) ( 4) = 2 4 + 3 12 = 2 12;
) ( 1) (6 ) = 6 2 6 + = 2 + 7 6;
) (5 + ) ( 2) = 5 10 2 2 = 2 7 10;
) ( 1) ( 7) = 2 + 7 + 7 = 2 + 6 + 7;
3) ) (5 7) (3 + 1) = 152 + 5 21 7 = 152 16 7;
) (3b + 7) (4 3b) = 12b 9b2 + 28 21b = 9b2 9b + 28;
) (2 3) ( + 2) = 22 + 4 3 62 = 22 + 62;
) (12 + 11) (10 5) = 120 602 110 55 =
= 602 175 110;
4) ) (52 + 1) (3 1) = 152 52 + 3 1;
) (52 + 1) (32 1) = 154 52 + 32 1 = 154 22 1;
) (a2 + b) (a b2) = a3 a2b2 + ab b3;
) (2 b) (a b2) = a3 a2b2 ab + b3;

114

5) ) ( + 3) (2 1) = 3 2 + 32 3 3 = 3 + 22 4 3;
) (7 1) (2 5 + 1) = 73 352 + 7 2 + 5 1 = 73 362 +
+ 12 1;
) (a + b 1) (b + a) = ab + a2 + b2 + ab b a = a2 + 2ab + b2
b a;
) (a + 3b) (a 3b 1) = a2 3ab a + 3ab 9b2 3b = a2 9b2
a 3b;
6) ) 5 ( + 2) ( + 3) = 52 + 15 + 10 + 30 = 52 + 25 + 30;
) 6 ( + 4) ( 1) = 62 + 6 24 + 24 = 62 18 + 24;
) (2 + 3) (5 1) = 102 2 + 153 32 = 153 + 72 2;
) 3b (b c) (c + 4b)=3b2c + 12b3 3bc2 12b2c = 12b3 9b2c 3bc2.
2. 1) ) (2 + 1) (2 + 1) = 4 3 + 2 + 3 2 + 2 + 1 =
= 4 2 + 2 1;
) (2m2 + 3m + 1) (2m2 + 3m 1) = 4m4 + 6m3 2m2 6m3 + 9m2
3m 2m2 + 3m 1 = 4m4 + 5m2 1;
2) ) ( 1) (4 3 + 2 + 1) = 5 4 + 3 2 + 4 + 3 2 +
+ 1 = 5 24 + 23 22 + 2 1;
) (4 + 2 5) (1 ) = 4 4 + 2 + 2 3 5 + 6 =
= 6 5 3 + 22 5 + 4;
3) ) ( + 5) ( 2) (2 3 10) = 4 33 102 + 33 92 30
102 + 30 + 100 = 4 292 + 100;
) ( 1) (2 + + 1) (6 + 3 + 1) = (3 1) (6 + 3 + 1) = 9 1.
3. (2 4b) (3a 8b) = (4b 2a) (8b 3a) = 2 (2b a) (8b 3a).
4. ) ( + 1) ( 3) = 2 2 3; ) ( 5) ( + 4) = 2 20.
34
1. 1) ) (2b 3) (5b + 7) + 21 = 10b2 + 14b 15b 21 + 21 = 10b2 b;
) 52 + (3 5) ( + 11) = 52 + 3 + 33 52 55 = 52 + 33;
2) ) 5 ( + 1) (4 + 1) = 5 42 4 1 = 42 1;
) 82 (3 1) (5 2)=82 (152 6 5 + 2) = 1204 883 + 162;
3) ) ( + 4) ( 3) (2 + 5) = 2 3 + 4 12 2 5 = 4 12;
) ( + 4) ( 3) ( + 7) = 2 + 4 2 7 + 3 + 21 = 21;
) (2 1) + ( + 3) ( 5) = 22 + 2 5 + 3 15 =
= 32 3 15;
) ( + 3) (3 + ) ( ) = + 32 32 + 3 + 2 =
= 3 + 2.
2. ) (5 + 1) (2 3) = (10 3) ( + 1);
102 15 + 2 3 = 102 + 10 3 3;
20 = 0; = 0;
) (7 1) ( + 5) = (3 + 7) ( + 3);
72 + 35 5 = 3 + 9 + 72 + 21;
10 = 14; = 1,4.

115

3. ) ( + ) (2 + 2) (2 ) = 2 + 2 23 + 2 22 + 3 =
= 23 + 3 + 22 2;
) (8a 3b) (3a 8b) (3a + 8b) (8a 3b) = 24a2 64ab + 9ab 24b2
24a2 + 9ab 64ab + 24b2 = 128ab + 18ab = 110ab;
) (p3 3k) (p2 + 3k) (p2 3k) (p3 + 3k) = p5 + 3kp3 3kp2 9k2 p5
3kp2 + 3kp3 + 9k2 = 6kp3 6kp2.
4. 1) at + (t 1) (a + 14) = at + at + 14t a 14 = 2at + 14t a 14;
2at + 14t a 14
3) Acp =
.
2) t + t 1 = 2t 1;
2t 1
5. ; + 1 ; ; b ;
S = ab ;
( 0,5) ( + 1 0,5) 6000 + 25500 = ( + 1) 6000;
60002 1500 + 25500 = 60002 + 6000; 6000 = 24000;
= 4 () ; 4 + 1 = 5 () .
35
1. 1) ) x (a b) + y (a b) = (a b) ( + );
) ( + ) b (x + c) = (x + c) (a b);
) 2 ( ) + ( ) = ( ) (2 + );
) 9 (a + b) (a + b) ab = (a + b) (9 ab);
2) ) b (a + 1) (a + 1) = (a + 1) (b 1);
) ( 3) ( 3) = ( ) (1 );
3) ) 5 (b 4) + x (4 b) = (b 4) (5 x);
) 2 ( 7) (7 ) = ( 7) (2 + );
4) ) ( 8) + (8 ) = ( 8) ( 1);
) + ( ) = ( ) (1 ).
2. 1) ) ( ) + b (x y) = (x y) (a + b);
) 5 ( + ) + ( + ) = ( + ) ( + 5);
2) ) 2 ( + ) + ( + ) = ( + ) ( + 2);
) 2 ( + 7) + ( + 7) = ( + 7) ( + 2);
3) ) a (b + c) 4 (b + c) = (b + c) (a 4);
) 3 (a m) y (a m) = (a m) (3 y).
3. 1) ) 2 + 3by + 6ay + bx = 2a (x + 3y) + b (x + 3y) = (x + 3y) (2a + b);
) 3 + 32 = 3 (1 + ) (1 + ) = ( + 1) (3 );
) 12bx + 3ax 4by = a (y + 3x) 4b (y + 3x) = (3x + y) (a 4b);
) a2b2 + ab + abc + c = ab (ab + 1) + c (ab + 1) = (ab + 1) (ab + c);
2) ) ax + bx + cx + ay + by + cy = x (a + b + c) + y (a + b + c) =
= (a + b + c) (x + y);
) ab a2b2 + a3b3 c + abc ca2b2 = ab (1 ab + a2b2) c (1 ab +
+ a2b2) = (1 ab + a2b2) (ab c);
3) ) xm + 1 xm + x 1 = xm (x 1) + x 1 = (x 1) (xm + 1);
) , .. .

116

4. ) 2 + 2 + 4 + 8 = ( + 2) + 4 ( + 2) = ( + 2) ( + 4);
) 2 8 + 15 = 2 3 5 + 15 = ( 3) 5 ( 3) = ( 3) ( 5).
36
1. 1) q2 p2;
2) (q + p) 2;
3) a3 + b3;

4) ( ) ( + );
5) m2 2mn.

2.

a2 + (2b) 2
92 + c2

(a + 2b) 2
(9 + c) 2
(0,3b + 1) 2

x2 y2
(7b) 2 22
(ac) 2 (3a) 2

(x y) 2
(5a 6b) 2

3.

c2 + (11b) 2
132 + (13b) 2

(8 + c) 2
(81a + 0,4) 2

0,12 (0,1a) 2
172 b2

(ab cd) 2
(7 x) 2

4. ) (2 + 2) (2 2); ) 2 (a + b) 2 (a b) 2.
37
1. 1) ) ( + 5) 2 = 2 + 10 + 25; ) (2 + ) 2 = 4 + 4 + 2;
) ( + ) 2 = 2 + 2 + 2;
2) ) ( 2) 2 = 2 4 + 4; ) (6 ) 2 = 36 12 + 2;
) ( 12) 2 = 2 24 + 144;
3) ) (5 2) 2 = 252 20 + 4; ) (2 + 9) 2 = 42 + 36 + 81;
) (6 1) 2 = 362 12 + 1;
4) ) (4 + ) 2 = 162 + 8 + 2; ) (7m 3n) 2 = 49m2 42mn + 9n2;
) (3 + ) 2 = 92 6 + 2;
5) ) (2 1) 2 = 4 22 + 1; ) (b + c3) 2 = b2 + 2bc3 + c6;
) (2 2) 2 = 4 222 + 4.
2.

4
b
0,2x
5
1
x
3
3
ab
2
2
2
a2b2
6

16a + 8ab + b
0,042 + 2 + 25
1
9 y 2 + 2 xy + x 2
9
a2b2 + 4ab + 4
4 + 42 + 42
4 4
a b + 12a2b2 + 36

4a2 8ab b2
0,042 2 + 25
1
9 y 2 2 xy + x 2
9
a2b2 4ab + 4
4 42 + 42
4 4
a b 12a2b2 + 36

3. 1) (a + (b + c)) 2 = 2 + 2 (b + c) + (b + c) 2 = a2 + 2ab + 2ac + b2 +

+ 2bc + c2;
2) (a (b c))2=a2 2a (b c) + (b c)2=a2 2ab + 2ac + b2 2bc + c2
3) (x + y + z) 2 = x2 + y2 + z2 + 2xy + 2xz + 2yz;
(x y + z) (x y + z) = (x y + z) 2 = x2 + y2 + z2 2xy + 2xz 2yz.

117

4.

1
(5 x 10 y )2 = 1 (5(x 2 y ))2 = 1 25(x 2 y )2 ;
25
25
25
25 ( 2) 2 = 52 ( 2) 2 = (5 ( 2)) 2 = (5 10) 2.

(x 2 y )2 =

38
1. 1) ) 2 + (5 3) 2 = 2 + 252 30 + 9 = 262 30 + 9;
) ( 2) 2 + 32 = 2 4 + 42 + 32 = 42 4 + 42;
) (3 7b) 2 42ab = 9a2 42ab + 49b2 42ab = 9a2 84ab + 49b2;
) 812 (9 + 7) 2 = 812 812 126 492 = 126 492;
2) ) ( 4) 2 + ( + 8) = 2 8 + 16 + 2 + 8 = 22 + 16;
) ( 7) + ( + 3) 2 = 2 7 + 2 + 6 + 9 = 22 + 9;
) ( 5) 2 ( 2) 5 = 2 10 + 25 52 + 10 = 42 + 25;
) (b + 4) b (b + 2) 2 = b2 + 4b b2 4b 4 = 4;
3) ) 3 ( + ) 2 = 32 + 6 + 32 ) 4 ( 2) 2 = 42 + 16 162;
) (2 1) 2 = 43 42 + ) a (3a + b) 2 = 93 62b ab2.
2. 1) ) (2 3) 2 + (3 + 2) 2 = 42 12 + 92 + 92 + 12 + 42 =
= 132 + 132;
) (5a+3b)2 (5a 3b)2=25a2 + 30ab + 9b2 25a2 + 30ab 9b2=60ab;
2) ) ((((a b) 2 + 2ab) 2 2a2b2) 2 2a4b4) 2 a16 b16 = (((a2 + b2) 2
2a2b2) 2 2a4b4) 2 a16 b16 = ((a4 + b4) 2 2a4b4) 2 a16 b16 =
= (a8 + b8) 2 a16 b16 = a16 + 2a8b8 + b16 a16 b16 = 2a8b8.
3. 1) (2 3b) 2 + (7a 9b) b = 4a2 12ab + 9b2 + 7ab 9b2 = 4a2 5ab =
= a (4a 5b);
2) (4 + 2) 2 (3 + 2) 2 = 162 + 16 + 4 92 12 4 = 72 + 4 =
= (7 + 4).
4. ;
( + 2) 2 = 2 + 20; 2 + 4 + 4 = 2 + 20; 4 = 16; = 4.
39
1. 1) ) a2 6ab + 9b2 = (a 3b) 2; ) 9a2 + 6ab + b2 = (3a + b) 2;
2

3
1
2
1

2) ) a b ; ) a 2 ab + b 2 = a b ;
2
4
3
2

3) ) 1 2ab + a2b2 = (1 ab) 2; ) a4 + 2a2b + b2 = (a2 + b) 2.

2. ) (4 + ) 2 = 162 + 8 + 2; ) 492 14 + 1 = (7 1) 2;
) 25 10 + 2 = (5 ) 2;
) 362 36ab + 9b2 = (6a 3b) 2;
81a2 36ab + 4b2 = (9a 2b) 2; 324a2 36ab + b2 = (18a b) 2.
3. ) 162 8ab + b2 = (4a b) 2; 49a2 14ab + b2 = (7a b) 2;

49a 2 8ab +
118

16 2
4
b = 7a b ;
49
7

1 2
1

x + xy + 4 y 2 = x + 2 y ; 252 + 20 + 42 = (5 + 2) 2;
16
4

25 x 2 + xy +

1 2
1
y = 5x + y .
100
10

40
1. 1) ) (b + 3) (b 3) = b2 9; ) (2 ) (2 + ) = 4 2;
) (k y) (k + y) = k2 y2;
2) ) (2 1) (2 + 1) = 42 1; ) (7 + 3) (7 3) = 492 9;
1
1
1

) 3 a 3 + a = 9 a 2 ;
5
5
25

3) ) ( + 3) ( 3) = 2 92;
) (2a b) (2a + b) = 4a2 b2;
) (8 + 4) (8 4) = 642 162;
4) ) (10a b) (b + 10a) = 100a2 b2;
) ( + 4) (4 ) = 16 2;
) (5b + 1) (1 5b) = 1 25b2.
2.

2
3
2b
0,5p
4
2
1
k
a
5
7

6
b2
c2

( 2) ( + 2)
(3a 2b) (3a + 2b)
(0,5 4) (0,5 + 4)
1 2
1
2
k a k + a
5
7
5
7

( 6) ( + 6)
(b2 c2) (b2 + c2)

2 42
9a2 4b2
0,252 162
4 2 1 2
k
a
25
49
22 36
b4 c4

3. 1) ) (5 + ) ( 5) = 2 252;
) (5 ) (5 + ) = 252 2;
) (5 ) ( + 5) = 2 252;
2) ) (3 2k2) (3c + 2k2) = 9c2 4k4;
) (4b3 + 2a) (2a 4b3) = 4a2 16b2;
) (32 1) (1 + 32) = 64 1;
3) ) (an 1) (an + 1) = a2n 1;
) (3n yn) (x3n + yn) = x6n y2n;
) (bn+2 cn2) (bn+2 + cn2) = b2n+4 c2n4;
4) ) ((x a) + b) ((x a) b) = (x a) 2 b2 = x2 2ax + a2 b2;
) ( 3) ( + 3) = ( ) 2 9 = 2 2 + 2 9;
5) ( + ) ( ) (2 + 2) (4 + 4) (8 + 8) = (2 2) (2 + 2) (4 +
+ 4) (8 + 8) = (4 4) (4 + 4) (8 + 8)=(8 8) (8 + 8)=16 16.

119

 41
2

1
1
1
1
1

1. ) 3 x y 3 x + y = 9 x 2 y 2 ; ) 5a = 25a 2 5a + ;
2
4
4
16
4

2 2
2 2
2
2
) (ab + xy) (ab xy) = a b x y ; ) (6+ 10) = 36 + 120 + 1002;
) (0,3b 3c) (0,3b + 3c) = 0,09b2 9c2; ) (ab + 7) 2 = a2b2 + 14ab + 49.
2. 1) ) (3 + ) (3 ) + 2 = 92 2 + 2 = 92;
) ( + 11) 2 20 = 2 + 22 + 121 20 = 2 + 2 + 121;
) 252 ( 5) ( + 5) = 252 2 + 252 = 502 2;
) 42 ( 3) 2 = 42 2 + 6 92 = 32 + 6 92;
2) ) (a + 2b) (a 2b) (a b) 2 = a2 4b2 a2 + 2ab b2 = 2ab 4ab;
) ( + ) 2 ( ) 2 = 2 + 2 + 2 2 + 2 2 = 4;
) (a 2b) 2 + (a + 2b) (a 2b) = a2 4ab + 4b2 + a2 4b2=2a2 4ab;
) ( 5)2 + ( + 5)2 = 2 10 +252 + 2 +10 +252=22 +502;
) (b 1) (b + 1) (a + 1) (a 1) = b2 1 a2 + 1 = b2 a2;
) (3 2) (3 + 2) + ( + 8) ( 8) = 92 4 + 2 64 = 102 68.
3. ) (3 + 3) ( ) = 3 ( + ) ( ) = 3 (2 2) = 32 32;
) (a b) (4a + 4b) = (a b) 4 (a + b) = 4 (a2 b2) = 4a2 4b2;
) (5 + 5) ( + ) = 5 ( + ) ( + ) = 5 ( + ) 2 = 52 + 10 + 52;
) (2 2) (3 3) = 2 ( ) 3 ( ) = 6 ( ) 2 = 62 12 + 62.
4. ) (1 5) (1 + 5) (3 1) 2 = 1 252 92 + 6 1 = 6 342 =
= 2 (3 17);
) (a + 2b) (2b a) + (a + 3b) 2 = 4b2 a2 + a2 + 6ab + 9b2 = 6ab +
+ 13b2 = b (6a + 13b).
5. 1) (2 1) (2 + 1) (22 + 1) (24 + 1) (28 + 1) (216 + 1) 232 = (22 1) (22 +
+ 1) (24 + 1) (28 + 1) (216 + 1) 232 = (24 1) (24 + 1) (28 + 1) (216 +
+ 1) 232 = (28 1) (28 + 1) (216 + 1) 232 = (216 1) (216 + 1) 232 =
= 232 1 232 = 1;
2) (2 + 1) (22 + 1) (24 + 1) (28 +1) 216=(2 1) (2 + 1) (22 + 1) (24 + 1)
216 = (2 1) (2 + 1) (22 + 1) (24 + 1) (28 + 1) 216 = (22 1) (22 +
+ 1) (24 + 1) (28 + 1) 216 = (24 1) (24 + 1) (28 + 1) 216 = (28
1) (28 + 1) 216 = 216 1 216 = 1.
42
1. 1) ) 92 4 = (3 2) (3 + 2); ) 1 252 = (1 5) (1 + 5);
) 36 492 = (6 7) (6 + 7);
1
25
5
5
1
1

2) )
c 2 = c + c ; )
+ y 2 = y y + ;
36
36
6
6
6
6

) 25 p 2
120

4
2
2

= 5 p 5 p + ;
121
11
11

3) ) 42 2 = (2 ) (2 + ); ) 16a2 b2 = (4a b) (4a + b);

) 81k2 c2 = (9k c) (9k + c);
4) ) 362 252 = (6 5) (6 + 5); ) 92 812=(3 9) (3 + 9);
) 492 642 = (7 8) (7 + 8);
5) ) a2b2 9 = (ab 3) (ab + 3); ) 2 22 = ( ) ( + );
) 24 1 = (2 1) (2 + 1).
2. 1) ) 81 6422 = (9 8) (9 + 8);
) 144422 225 = (122 15) (122 + 15);
2) ) (2a + 7b) 2 (3a 5b)2 = (2a + 7b 3a + 5b) (2a + 7b + 3a 5b) =
= (12b a) (2b + 5a);
) ( + ) 2 ( ) 2 = ( + + + ) ( + +
) = 2 (2 2) = 4 ( );
3) ) a2n 1 = (an 1) (an + 1);
) x2 y4n = (x y2n) (x + y2n);
) a4n b4n = (a2n b2n) (a2n + b2n);
) 49x4n 25 = (7x2n 5) (7x2n + 5);
4) ) (3 19) + (3 4) (2 + 9) = 32 19 + 62 + 27 8 36 =
= 92 36 = (3 6) (3 + 6);
) (5 4) (3 + 4) 4 (3,5 + 2) = 152 + 20 12 16 142
8 = 2 16 = ( 4) ( + 4).
3. 2n 2n + 2 ;
(2n + 2) 2 (2n) 2 = (2n + 2 2n) (2n + 2 + 2n) = 2 (2n + (2n + 2));
;
(2n + 3) 2 (2n + 1) 2 = (2n + 3 2n 1) (2n + 3 + 2n + 1) =
= 2 ((2n + 1) + (2n + 3)).
C 43
1. 1) ) (2 + ) ( 3) + (2 ) = 22 6 + 32 + 2 2 =
= 2 3 32;
) (3 + ) ( + ) 4 ( ) = 32 + 3 + + 2 4 + 42 =
= 32 + 52;
) 2b (b + 4) + (b 3) (b 4) = 2b2 + 8b + b2 4b 3b + 12 = 3b2 +
+ b + 12;
) 3 ( 5) ( 4) ( + 8) = 32 15 2 8 + 4 + 32 =
= 22 19 + 32;
2) ) (2x b) (3x + b) + (3b x) (b + x) = 6x2 + 2bx 3bx b2 + 3b2 +
+ 3bx bx x2 = 5x2 + bx + 2b2;
) ( + 2) ( 3) ( + 1) ( + 3) = 2 3 + 2 6 2 3 3 =
= 5 9;
) ( 10) ( 2) + ( + 4) ( 5) = 2 2 10 + 20 + 2 5 +
+ 4 20 = 22 13;
) ( 5) ( + 1) ( 6) ( 1) = 2 + 5 1 2 + + 6 6 =
= 3 7.

121

2. 1) ) ( 4) ( + 4) 2 (3 ) = 2 16 6 + 22 = 32 6 16
) (4 3) 2 6 (4 ) = 162 24 + 9 24 + 62 = 222 48 + 9
2) ) ( 8) ( 7) ( 9) 2 = 2 7 8 + 56 2 + 18 81 =
= 3 25;
) ( + 3) ( 11) + ( + 6) 2 = 2 11 + 3 33 + 2 + 12 + 36 =
= 22 + 4 + 3;
3) ) (b + 3) (b 3) + (2b + 3) 2 = b2 9 + 4b2 + 12b + 9 = 5b2 + 12b;
) ( ) 2 + ( + ) 2 = 2 2 + 2 + 2 + 2 + 2 = 22 + 22;
4) ) 3 ( 5)2 +(10 82)=32 30 + 75 + 10 82=52 20 + 75;
) 2 ( + 6) 2 (20 + 70) = 22 + 24 + 72 20 70 = 22 + 4 + 2.
3. ) (2 + 3) (5 ) (2 3) (5 + ) = 10 2 + 15 32 10 2 +
+ 15 + 32 = 26; 26 (1,1) = 28,6;
) (3a + b) 2 (3a b) 2 = 9a2 + 6ab + b2 9a2 + 6ab b2 = 12ab;
10
12 ( 0,3) = 12 .
3
4. 1) ) 8 (5 + 3) 2 + 9 (3 1) 2 = 2002 + 240 + 72 + 812 54 + 9 =
= 2812 + 186 + 81;
) 7 (2 5) 2 2 (7 1) 2 = 282 140 + 175 982 28 2 =
= 702 168 + 173;
2) ) (42 + 3)2 + (9 42)2 2 (42 + 3) (42 9) = (42 + 3 + 9 42)2=
= 144;
) (a2 6ab + 9b2) (a2 + 6ab + b2) (a2 9b2)2=(a 3b) 2 (a2 + 6ab +
+ b2) (a 3b) 2 (a + 3b) 2 = (a 3b) 2 (a2 + 6ab + b2 (a + 3b) 2) =
= (a2 6ab + 9b2) (a2 + 6ab + b2 a2 6ab 9b2) = (a2 6ab +
+ 9b2) (8b2) = 8a2b2 + 48ab3 72b4;
3) ) (x + 3b) (x 3b) (x + 2b) (x2 2bx + 4b2) = x2 9b2 x3 8b3
) ( + 1) (2 + 1) ( 1) (2 1) = 3 + 2 + 2 +
1 3 + 2 + + 2 1 = 42 2.
5. 1) ( 3) ( + 3) + (3 ) (3 + ) + ( ) ( + ) = 0;
2 92 + 92 2 + 2 2 = 0;
2) (a b) (a + b) ((a b) 2 + (a + b) 2) = 2 (a4 b4);
(a2 b2) (a2 + b2 2ab + a2 + b2 + 2ab) = (a2 b2 (2a2 + 2b2) =
= 2 (a2 b2) (a2 + b2) = 2 (a4 b4).
C 44
1. 1) ) 52 45 = 5 (2 9) = 5 ( 3) ( + 3);
) 2 4 = (2 4) = ( 2) ( + 2);
) 18 22 = 2 (9 2) = 2 (3 ) (3 + );
) 3ky2 3k = 3k (y2 1) = 3k (y 1) (y + 1);
2) ) 32 752 = 3 (2 252) = 3 ( 5) ( + 5);
) 22 + 23 = 2 (2 2) = 2 ( ) ( + );
) 53 52 = 5 (2 2) = 5 ( ) ( + );
) bc3 b3c = bc (c2 b2) = bc (c b) (c + b).

122

2. 1) ) 5a2 + 10ab + 5b2 = 5 (a2 + 2ab + b2) = 5 (a + b) 2;

) 4 + 4 = ( 4 + 4) = (4 3);
) 2 2 + 2 = ( ) 2;
) 3 + 22 + = (2 + 2 + 1) = ( + 1) 2;
2) ) 62 + 12ab 6b2 = 6 (a2 2ab + b2) = 6 (a b) 2;
) 22 8 8 = 2 (2 + 4 + 4) = 2 ( + 2) 2;
) 2 + 8ab 16b2 = (a2 8ab + 16b2) = (a 4b) 2;
) 123 + 122 3 = 3 (42 4 + 1) = 3 (2 1) 2.

1 2
1
1
a + ab + b 2 = (a + b )2 ;
2
2
2
1 3
1
1 3
1
1

) a 3 = 3 a 1 = 3 a 1 a 2 + a + 1 ;
9
27
3
9
3

3. 1) )

2) ) 4 82 + 16 = (2 4) 2 = ( 2) 2 ( + 2) 2;
) + 7 = (6 1) = (3 1) (3 + 1) =
= ( 1) (2 + + 1) ( + 1) (2 + 1);
3) ) ( + 5) 2 ( + 5) 2 + ( + 5) = ( + 5) (2 2 + 1) =
= ( + 5) ( 1) 2;
) 4 2 2 (4 2) + 2 (4 2) = (4 2) (1 2 + 2) = (2
) (2 + ) (1 ) 2;
4) ) 8a3 b3 + 4a2 + 2ab + b2 = (2a b) (4a2 + 2ab + b2) + 4a2 + 2ab +
+ b2 = (4a2 + 2ab + b2) (2a b + 1);
) 8a3 b3 + 4a2 4ab + b2 = (2a b) (4a2 + 2ab + b2) + (2a b) 2 =
= (2a b) (4a2 + 2ab + b2 + 2a b).
4. 1) ( + 1) 3 ( + 1) = ( + 1) (( + 1) 2 1) = ( + 1) (2 + 2 + 1 1) =
= ( + 1) ( + 2);
2) 4b2c2 (b2 + c2 + a2) 2 = (2bc b2 c2 a2) (2bc + b2 + c2 + a2) =
= (a2 (b c) 2) (a2 + (b + c) 2) = a4 a2 (b + c) 2 a2 (b c) 2
(b c) 2 (b + c) 2;
(a + b + c) (a b + c) (a + b c) (b + c a) = ((b + c) + a) ((b + c)
a) (a + (b c)) (a (b c)) = ((b + c) 2 a2) (a2 (b c) 2) =
= a2 (b + c) 2 (b + c) 2 (b c) 2 a4 + a2 (b c) 2;
,
4b2c2 (b2 + c2 + a2) 2 (a + b + c) (a b + c) (a + b c) (b + c a);
, .
, :
4b2c2 (b2 + c2 a2) 2 = (a + b + c) (a b + c) (a + b c) (b + c a);
(2bc b2 c2 + a2) (2bc + b2 + c2 a2) = (a2 (b c) 2) ((b + c) 2 a2).
5. 1) ( 1) ( 3) = 2 3 + 3 = 2 4 + 3;
2) (2 4 + 3) ( + 1) = 3 + 2 42 4 + 3 + 3 = 3 32 + 3.

123

C 45
1. 1) )

)
M(-3; 9)

6
4

=6-

1
0

-3

2) )

-6

(5; -6)

= 3 3
y = 2

=5

2
3

1
0

(2; 3)

1 2

-3

2. . 28
(2; 3)
y = 2 0,5 x
;

y = x + 5

3 = 2 0,5( 2 ) = 3
;

3 = 2 + 5 = 3

124

= -3

= 4 2

= -6

1
-6

(-6; 6)

. 28
(4; 2)
y = 4 1,5 x
;

y = x 6
2 = 4 1,5 4 = 2
.

2 = 4 6 = 2

= -

3. 1) )

=+4

= -
4

2
1
-4

-2

(-2; 2)

= - + 3
=1
3

1
0

-1

(2; 1)

2) )

= 5 2

=
y=

x
2

1
1

1
-2

0
-1

0
-1
-2

-2
(-2; -2)

(3,5; -2)

125

) (4,7; -0,8)

4.

) (2,5; 0,7)

=4

y = 2

= 1,5 3

2
0

1
1

y=

-2

-3

-3

) (2,2; 0,4).
y = 1,5

x
2

= 2 4

1
0

-4

5. , k1 k2.
, k1 = k2, b1 b2.
, k1 = k2, b1 = b2;
y = 5x 7
. : m 5, .. m ,
1)
y = mx + 3

5. : m = 5;
: m .
y = 0,5 x + m

2)
5 . : m . :
y = 1,5 x 4

m . : m .
m

y = x 2
. : m 6, m , 6
3)
3
y = 2x 2

: m . : m = 6.
126

 46
1. 1) ) = 7 ; = 7 ; ) = 2; = + 2; ) = ; = ;
1
2) ) = 2 8; y = x + 4 ;
2
1
) = 3; y = x ;
3
1
) = 2 + 5; x = y 2,5 ;
2
2
3
3) ) x = y ; y = x ;
3
2
) = 2,5 5; = 0,4 2;
8
3
) x = y 0,8 ; y = x 0,3 .
3
8

x = 7 y
x = 1 1 + 6 = 7
;
2. 1) )
;
;
14 2 y + y = 8 y = 6 2 1 + 6 = 8
x = y 2
x = 8 8 + 6 = 2
)
;
;
;
y 2 2 y = 4 y = 6 8 + 12 = 4
x = y
x = 2 2 2 = 0
)
;
;
;
+
=
3
y
y
8

y = 2 6 + 2 = 8
x = 2 y 8
x = 8 8 0 = 8
;
)
;
;

2
y
8
3
y
8

y = 0 8 0 = 8
x = 2 y 3
x = 1 3 + 4 = 7
2) )
;
;
;
6 y 9 + 2 y = 7 y = 2 1 4 = 3
m = 3n + 8
m = 2 2 + 6 = 8
)
;
;
;
6n + 16 3n = 10 n = 2 4 + 6 = 10
b = 10 2a
b = 2 20 6 = 14
;
)
;
;

+
=
5
a
30
6
a
14

a = 4 8 + 2 = 10
c = 2 p + 5
c = 3 3 + 2 = 5
)
.
;
;
+

=
4
p
10
3
p
9

p = 1 6 + 3 = 9
3.

x y = 12
)
;
x + y = 8
x y = 11
)
;
x + y = 9

x = y + 12
x = 10
;
;

y + 12 + y = 8 y = 2
x = y + 111
x = 60
;
;

y + 111 + y = 9 y = 51

127

 x + y z = 1 2 y + y y 3 = 1

4. 1) z = y + 3 ; z = y + 3
;
x = 2 y
x = 2 y

x = y + 1
x = y + 1

2) y z = 2
; y = z + 2
z y 1 = 3 z z 2 1 = 3

y = 2

z = 5 ;
x = 4

x = y + 1

y = z + 2 ;
3 = 3

y = z + 2

.
x = z + 3
z

47

2 x 2 y = 6
1. 1) )
;
3 x + 2 y = 1

5 x = 7
;

x y = 3

6 p 2c = 4
)
;
3 p + 2c = 6

9 p = 10
;

3 p c = 2

2a 2b = 8 5b = 6
)
;
;
2 z + 7b = 2
a + b = 4
15 z 21x = 9
4a + 6b = 2 8b = 1
;
)
;
2) )
4
+
2
=
3
2

3
=
1
a
b
a
b

15 z + 25 x = 10
12 x 16 y = 40 7 y = 25
)
;
.
12 x + 9 y = 15
3 x + 4 y = 10
2 x = 8
2. 1) )
;
x y = 3
2 a = 8
)
;
a b = 6

x = 4
;

y = 1

6 z = 12
x = 2
)
;
;
3 z + t = 8 t = 2

a = 4
;

b = 2

2u + 2v = 20 5v = 35
v = 7
2) )
;
;
;
+
=

2
u
3
v
15
u
v
10

u = 3
10 x + 5 y = 25 13x = 26
x = 2
;
;
;
)
3 x 5 y = 1
y = 5 2x y = 1
2m n = 1 3n = 3
n = 1
)
;
;
;
2 m 2 n = 4
2 m 2n = 4 m = 1
6a 4b = 2 11b = 22
b = 2
;
3) )
;
;
+
=
a
=

b
6
a
15
b
24
2
8
5

a = 1
9u 6v = 36 17u = 34
u = 2
;
;
;
)
+
=

8
u
6
v
2
2
v
3
u
12

v = 3
9 x + 6 y = 0 x = 38
x = 38
)
;
;
.
10 x 6 y = 38 3 x 2 y = 0 y = 57

128

4 x = 1

3 z 5 x = 2

x 2 + y 2 = 8
x + y = 12 4 x = 28
x = 7
3. 1)
;
;
;
;

=
+
=
3
(
x
2
)
(
y
2
)
12
3
x
y
16
x
y
12

y = 5
3(3a + 1) + 5(2b 1) = 6 9a + 10b = 8 9a + 10b = 8
2)
;
;
;
2(3a 2 ) + b 3 = 4
6a + b = 11 60a 10b = 110
51a = 102 a = 2
;
.

b = 11 6a
b = 1
y x z = 2 x + y = 3 x = 1

4. 1) x + y = 3
; x + z = 4 ; z = 3 ;
x + z = 4
y = 2
y = 2

x y z = 0 y = x z y = 3

2) 2 x 2 z = 6 ; z = x 3 ; z = 1 .
2 x = 8
x = 4
x = 4

48

y = 20 7 x
y = 20 7 x y = 1
1. 1) )
;
;
;
x 100 + 35 x = 8 36 x = 108
x = 3
x = 4 2 y
x = 4 2 y x = 17
)
;
;
;

+
=

20
10
y
8
y
1

2 y = 21
y = 10,5
10 x + 4 y = 0
21y = 105 y = 5
;
;
;
2) )

10
x
25
y
105

2 x = 5 y 21 x = 2

15 x 35 y = 25
)
15 x + 12 y = 21 ;

23 y = 46 ;
3x = 7 y 5

y = 2
;

x = 3

2 4 y = 3x 6
3x + 4 y = 8
3) )
2 x + 2 y = 5 y + 2,5 ; 2 x 3 y = 2,5 ;

6 x 8 y = 16 ; 17 y = 8,5 ; y = 0,5 ;
6 x 9 y = 7,5
2 x = 3 y + 2,5 x = 2

3x 6 y 2 = 2 x = 2 y
x = 2 y
x = 4
)
2 x + 2 1 = 3 y 1 ; 2 x 3 y = 2 ; 4 y 3 y = 2 ; y = 2

y = 2x 4
2. )
;
y = 3 x + 1

y = 2x 4
2 x 4 = 3x + 1 ;

4 x 3 y = 1 8 x 6 y = 2 ;
)
3x + 2 y = 12 ;

9 x + 6 y = 36

y = 2 ;
x = 1

17 x = 34
x = 2
2 y = 12 3 x ; y = 3 .

129

1
2
3 x = 2 + y 2
;
3. )
2 x = 8 y
3

2
x = 8 y
;
3
2 + 0,5 y = 8 y

y = 4
x = 3 (4 ) = 6 ;

6a + 3b = 36 ; b = 8a 8
)

b=8.
;

6a + 24a 24 = 36 a = 2

8a b = 8

x=
a = 5 ;
5 ;
b = 3
1

y =
3

2a b = 5 4a = 12
a = 3 x = ;
1)
2a + b = 7 ; b = 7 2a ;; b = 1 ;
3

y = 1
3a + 5b = 11 24a 40b = 88 ;
2)
8a 7b = 9 ; 24a 21b = 27

a b = 8
4.
2a + 3b = 1 ;

a = b + 8
2b + 16 + 3b = 1 ;

b = 1 ; x = ;
a = 2
2

y = 1
a +b =1
a = 1 b
3)
0,5a + 2b = 8 ; 0,5 0,5b + 2b = 8 ;

x=

a
b
a
=
1

4.
1,5b = 7,5 ; b = 5 ;
1

y =
5

61b = 61 ;
3a = 11 5b

49

x + y = 36
x + y = 36
x + y = 81
1. 1) )

;
x y = 15 ; )

x = 2 y
y = 2x
x + y = 35
y x = 2
2) )
; )
;
x = 1,5 y
2 y + x = 17
3 x + 2 y = 1200
3 x + 2 y = 172
3) )
; )
.
y
=
x
+
100

x + 4 y = 198
2. 1) 30, 4
.
2) 65 , 5 .

130

a + b
2 = 36
a b = 140
3. 1)
; 2)
0,6a 0,7b = 64 ; 3)
1

(a b ) = 0,8
5

x + y + z = 10
.
x = 2 y
x z = 5

50
1) ;
-;

x + y = 700 ; 2 x = 1000 ; x = 500 ;

x y = 300 y = 700 x y = 200

2)

x + y = 8
10 x + 5 y = 65 ;

x = 8 y
x = 5
80 10 y + 5 y = 65 ; y = 3 ().

3)

5 x + y = 25 ;
5 x y = 15

10 x = 40 ; x = 4 ( )
y = 25 5 x

y = 5 ( )

4)

3x + 2 y = 24 ; 3x + 2 x + 4 = 24 ; x = 4 (/).
y x = 2
y = x + 2
y = 6

5) - ; -

x + y = 20
x 4 = 2( y 4) ;

x = 20 y ;
x 2 y = 4

x = 20 y
x = 12
20 y 2 y = 4 ; y = 8 ().

6) ;

x y = 60
x = y + 60
x = y + 60 x = 300
0,6 x + 12 = 0,8 y ; 0,6 y + 36 + 12 = 0,8 y ; 0,2 y = 48 ; y = 240

7) ();
;

3(x y ) 2(x + y ) = 5 ; 3x 3 y 2 x 2 y = 5
(x y ) = 0,75(x + y ) x y 0,75 x 0,75 y = 0 ;

7
y

5 y = 5 ; y = 2,5 ;
x 5 y = 5

0,25 x 1,75 y = 0 ; x = 7 y
x = 17,5

( + ) ; ( ) ;
S = 3 ( ) + 2 ( + ) = 3 15 + 2 20 = 45 + 40 = 85 .
: 85 .
131

 51

4
1
5 0,25 1
3 1
= ;
= 0,5 ; = ;
1. ) 9
2
18
6 2
1,5
2,5 + 0,4
0,5 2
= 2,9 .
)
= 2,5 ;
2,5 0,4
1

2.

3
x +1
2x
x +1

1
2

2
5

1,5

3
2
1
2

2
1
0

3
4
1

3. ) 0;
) 5 0; 5;

) 3b + 1 0; b

1
;
3

) 4 0 , ;
) 2 + 4 0 , .. 2 0, .

b 3 0 ; b 3 , .. b 3 b 6;
4. ) (b 3) (b 6) 0, ..

b 6 0 b 6

) ( 5) ( + 5) 0, .. 5 5.
52

5
a
m
p
; ) ; )
; )
;
7
q
n
n
p( p + c )
p
=
2) )
;
( p c )( p + c ) p c
p( p c )
pc
;
)
=
p( p + 2c ) p + 2c

1. 1) )

3) )
)

)
)

p
;
pc
p+c
;
p

a2
a
=
;
a(a + 3) a + 3
a 2 + 3c
, , , a2 9

a 2 + 3a
a(a + 3)
a
=
=
;
2
(a 3)(a + 3) a 3
a 9

132

p( p + c )
=
( p c )( p + c )
( p c )( p + c ) =
p( p c )

a2
a
a(a 3)
a
=
=
;
)
;
(a 3)(a + 3) a + 3
a(a 3) a 3
1
;
) ;
4) )
y
x 2y
1
(x 3 y )(x + 3 y ) = x 3 y ;
=
; )
)
(x 2 y )(x + 2 y ) x + 2 y
x + 3y
)

(a 5b )2

= a 5b ;

= x + 4y ;
x + 4y
3(x 2 y )
3
6) )
= ;
4(x 2 y )
4

5) )

a 5b

(x + 4 y )2

) 3b;

2. )

)
3. )

(57 38)(57 + 38) 19 95

=
(22 3)(22 + 3) 19 25
(83 + 17 )2 = 100 2 = 100 .
100

( x 3 y )2 = ( x 3 y ) ;
(x 3 y )(x + 3 y ) (x + 3 y )
(3x + y )2 = 3x + y ;
(3x y )(3x + y ) 3x y
(x 4 y )(x + 4 y ) = x 4 y ;
4y + x

(x 4 y )(x + 4 y ) = (x + 4 y ) .
(x 4 y )

19 5
55

=3

4
5

100

a b + x(a b ) (a b )(x + 1) x + 1
=
=
;
a b + y (a b ) (a b )( y + 1) y + 1

a b 0, .. a b; 5,17 7,15 . :
)

c a + (c a )(c + a )
c + a + (c + a )

(c a )(c + a + 1) = c a ;
(c + a )(c + a + 1) c + a

1,3 + 1
= 1 ;
3,3 + 1

+ + 1 0; 2,73 2,74 + 1 0 .
:

2,73 + 2,74
5,47
=
= 547 .
2,73 2,74 0,01

53

3 5 3+5 8
+ =
= ;
b b
b
b
8a 3a 5a a
)
=
= ;
5x
5x x

1. 1) )

3x 7 + 5 x + 7 8 x 8
= 2 = ;
x
x2
x
2
2
2
9a 3b 2a + 3b 7 a
)
=
= 7a ;
a
a
)

133

x3
;
x4
y 2 25 ( y 5)( y + 5)
)
=
= y+5;
y 5
y 5

2) )

y 2 10 y + 25 ( y 5)
=
= y 5 ;
y 5
y 5

b 2 1 (b 1)(b + 1) b + 1
.
=
=
a(b 1)
a(b 1)
a

8 3 + 11 16
=
;
b
b
3 x + 5 + 7 x 11 9 x + 3 x 3
=
= 1;
2)
x3
x3
3x + 5 2 x 7
1
x2
=
= 2
.
3)
2
x 4
x 4 x+2

2. 1)

3. 1) )

16a 2 8 + 1 16a 2 7
;
=
16a 2 1
16a 2 1
8a 3 + 3a 1 + 5a 7 8a + 9

(2a + 1)2

(2a + 1)(4a 2 2a + 1) = ;
(2a + 1)2

4a 2 2a + 1
;
2a + 1
5
1
c 8
c3
=
+
=
;
2) )
(c 3)(c 7 ) (c 3)(c 7 ) (c 3)(c 7 ) c 7
=

x 2 6x + 9
(x 3) = (x 3) = 3 x ;
=
x3
x3
2
2
9b + 6b + 1 (3b + 1)
)
=
= 3b + 1 .
1 + 3b
3b + 1
2

54
1. 1) )

134

2(a 1) + 3a
5a 2
5(x y ) 2(x + y ) 3 x 7 y
=
; )
;
= 2
a(a 1)
a(a 1)
(x + y )(x y )
x y2

ab b(a b )
b2
p2 3p + 3p 6
p2 6
=
=
; )
;
a(a b )
a(a b )
( p 2)( p 3) ( p 2)( p 3)

a 2 2ac + c 2 a 2 2ac c 2
4ac
4ac
=
;
= 2
a2 c2
a c2 c2 a2
x 2 + 5x + 6 x 2 2 x 1
3x + 5
=
;
)
(x + 1)(x + 3)
(x + 1)(x + 3)
)

ax
x 2 x(x + a ) x 2 x 2 ax
=
= 2
;
2
2
2
2
x a
x a
a x2
4(a b ) + 1 4a 4b + 1
3+ 52
13
=
)
=
; )
;
4a 8
4a 8
a2 b2
a2 b2
4 2 3 1
5
n 2m
2(n m ) n
=
; )
=
;
)
6(x + 3)
6(x + 3)
m(n m )
m(n m )

2) )

a 2 a(a b )
ab
=
;
5(a b )
5(a b )

a(a + b ) ab
a2
3c + 1 + (c 1)(c 2 ) c 2 + 3
=
=
;
)
;
a+b
a+b
c 1
c 1
2
2(x 3) + x + 4 3x 2
a 2 2ac + c 2 (a c )
=
; )
;
)
=
x3
x3
ac
ac
( y + 2)( y 2) 2 = y 2 6 ; ) p 2 + k 2 + 2 pk = ( p + k )2 .
)
pk
pk
y+2
y+2

3) )

2. 1) )

x 7 x + 3y 3y 7
;
=
xy
xy

(2a + 1)6a 9a(a 5) + a + 4 = 12a 2 + 6a 9a 2 + 45a + a + 4 = ;

18a 2
3a + 52a + 4
;
=
18a 2

18a 2

4 2(c 5) + (c + 2 )(c + 5) 4 2c + 10 + c 2 + 7c + 10
=;
=
c 2 25
c 2 25
c 2 + 5c + 24
;
=
c 2 25
a(a + x ) 2a 2 + a(a x ) a 2 + ax 2a 2 + a 2 ax
=;
)
=
a2 x2
a2 x2
0
= 2
= 0.
a x2

2) )

135

3. 1) )

(x + 2 y )(x 2 2 xy + 4 y 2 )+ (x 2 y )(x 2 + 2 xy + y 2 ) = ;
(x 2 y )(x + 2 y )

x 3 + 8 y 3 + x 3 + 2 x 2 y + xy 2 2 x 2 y 4 xy 2 2 y 3
=;
(x 2 y )(x + 2 y )

2 x 3 + 6 y 3 3xy 2
;
(x + 2 y )(x 2 y )

=
2) )
=

=
=

4. 1) )

)
2) )

(x + 2 y )(x 2 2 xy + 4 y 2 ) (x 2 y )(x 2 + 2 xy + 4 y 2 ) = ;
(x 2 y )(x + 2 y )

x3 + 8y3 x3 + 8y3
x2 4y2

16 y 3
x2 4y2

(a b )2 a 2 b 2 + (a + b )2
a2 b2

=;

a 2 2ab + b 2 a 2 b 2 + a 2 + 2ab + b 2
a2 b2

2 y( y + 2) 2 y( y 2) + 2 y 2 + y 2 + 4
=;
2 y ( y 2 )( y + 2 )

2y 2 4y 2y 2 + 4y + 2y 2 + y 2 + 4
2y3 8y

(2 y )(2 + y ) = 1 .
2 y ( y 2 )( y + 2 )
2y

136

a2 b2

y2 + 4
2y3 8y

=;

x+2x+2
1
1
4

=
= 2
;
x2 x+2
x2 4
x 4

5(x + 6 ) 4(x 4 )
5
4
x + 46
= 2
;

=
2
x4 x+6
x + 2 x 24
x + 2 x 24

5 x 10 x + 7 10 x 2 5 x 10 x 2 30 x 7 x 21

=
=;
x + 3 2x 1
2 x 2 + 5x 3

42 x 21
;
2 x 2 + 5x 3
= 42; b = 21.
:
22 5 3 ;
(2 1) ( + 3)=22 + 5 3 .

a2 + b2

a
b
ax + 2a + bx 2b (a + b )x + 2a 2b
;
+
=
=
x2 x+2
x2 4
x2 4

a + b = 0
:
;
2a 2b = 1

a = b
;

2b 2b = 1

a = 4
.

b = 1
4

55

x 8y
2

= ; )
4 y 11x 11
7
;
2) )
8p
(a 2b )3(x + c ) =
)
(x + c )5(a 2b )

1. 1) )

3) )
4) )
)
)
)
5) )
)
)
6) )
)

5c 3 x 15a
a 9 10b10 5b 2
;
)
= 75 ;

=
ac 3 x
8b 8 a 11
4a 2
a(a 2b ) x + y
a
)
= ;

x(x + y ) a 2b x
3
b(a 2b ) 5( y + x )
5
= ;
;
)

y (x + y ) b(2b a )
y
5

1
a 2b
c
3a 35b 5
b8
b
7a 15

= ; ) 7
=
;
)
2 = ;
16
3a
7b 6a
2
9c 9a b 81
b
21a
8a + 3b
3c
c

= ;
3p
3b + 8a p
a 2b 3n + m

= 1 ;
m + 3n 2b a
(k c )(k + c ) (a b )(a + b ) = (k c )(a + b ) ;
k +c
a b
(a b )(a + b ) (a p ) = a + b ;
a b
a p
(a p )2
5a + 3 x
3
15
9
3 x = (5a + 3 x ) = a + x ;
2x
2
2
2
8 y 2 + 8 yz
;
y2 z
(2 x y )(5 x + y ) = 5 x + y ;
x(2 x y )
x
2b a + b 2a + 2b
a
a
=
; )

=
;
1 3b 2
3b
2(5b 3c ) 10b 6c
(m 2n )(m + 2n ) 1 = m + 2n .
(m 2n ) m + n
m+n
137

2. )

12 x 3 y 3x 2
;
=
8 xy 3
2y2

x
3y 2

x
x 2y
2
: = 2
=
;
x
3y
2y 3y

x 2x x 2 2x 4
;

=
3y 2 y y 2 3y5

x
3y 2

y y2
x
y2
y

2 = 2 3 = 2 ;
6x
2x x 3 y 2x

( p 2c )( p + 2c ) (a 5b )(a + 5b ) = ( p + 2c )(5b a ) ;
2c p
a + 5b
(a + 5b )2
3(a b ) (c + y )(c y 1) 3(c y 1)
;

=
4(c + y ) (a b )(a + b + 1) 4(a + b + 1)
a 3 b 3 (a b )(a + b )
ab
ab
;

=
2
3
3(a + b ) 18
a (a b )6ab
( y + 2)2 y( y + 1) 6( y 3) = 3( y + 2) = 3 y + 6 .
2( y + 1) ( y 3)2 y ( y + 2 )
y 3
y 3

3. 1) )

)
2) )
)

56

x2 y2
xy

1. 1) )

5 xy
(x y )(x + y )5 xy = 5 x + 5 y ;

=
xy(x y )
x y

3ab + 7b 2 + 8a 2 3ab 7b 2 + 8a 2
7b 2 + 8a 2

=
5ab
10ab
50a 2 b 2
2y2 b2 + y2 b + y
3y 2 b2
)
;
=
2
y( y + b)
b + y2
y b2 + y2

x + 2 xy + y
y
(x + y ) = x + y ;

=
2
x + y y (x + y )
y
y
y
2y x
2y x
)

=
;
y
2y + x 2y + x
p+ p2
2 p
2p 2 2 2p
)
.

=
=
2
2
p2
2p
2p
2p p + p
2

2) )

2. 1) ) 5

138

1
5a + 2b
50a 2 20ab 5a + 2b 5a 2b

=
=;
2a 2a(5a 2b )
2a(5a 2b )

50a 2 20ab 10a 50a 20b 10 25a 10b 5

;
=
=
2a(5a 2b )
2(5a 2b )
5a 2b

(5a 2b )10a = ;
10a 5a 2b 5a 2b + 5a + 2b

=
(5a + 2b )(5a 2b ) 2a(5a + 2b )(5a 2b )
2a
5
;
=
5a + 2b
1
5a + 2b
6
1 12a 5a 2b
)

=
=;
+ 1 =
5
5a + 2b
2a
2a(5a + 2b )
5a + 2b 2a
7 a 2b
=
;
2a(5a + 2b )
)

2) )

4a 2 2a + 1 3 + 6a + 3 4a 2 + 2a 4a + 1

=;
2a + 1
(2a + 1) 4a 2 2a + 1

(2a + 1) (4a 2a + 1) = 1 ;
(2a + 1)2 (4a 2 2a + 1)
(x 2 y )(x + 2 y )
x 3 8 y 3
1
+

)
2

x 2y

2 xy

(x 2 y )(x + 2 y ) x 2 + 2 xy + 4 y 2
=
+

2 xy
2 xy

3.

(2x

+ 2 xy 2 y

(2 xy )(x 2 y )

1 y 2 ( x y )2

x ( x + y )2

x(x + y )

2y

=;
x 2y

2x + 2 y
.
x 2y

xy + y 2 + x 2 y 2

y ( x y )2

y 2 (x y )2

2y
=;
x 2y

2 xy

x(x + y )

y (x y )

+ x =;
x+ y

y
x+ y
x
x
=
+
=
= 1.
x+ y x+ y x+ y x+ y

1
1
1. 6 x 8 y = 6

2
5
8 = 4 5 = 1 .
3
8

2. 0,8 6 1 = 5,8 < 3,8 = 0,8 6 1.

139

3. ) 2 3 11 + 8 = 9 + 5;
) 5 (2 + 1) 3 = 10a + 5 3 = 10 + 2;
) 14 ( 1) + (2 + 6) = 14 + 1 + 2 + 6 = 15 + 7.
4. 4 (2,5 1,5) + 5,5 8 = 10 + 6 + 5,5 8 = 4,5 2;

45 2
2
2 = 1 .
4,5 2 =
10 9
9
5. /.
: V t + x t = 5;
S Vt
x=
;
t
200 60 2
x=
= 40 /.
2
6. 3 (5 (3 1)) = 3 5 + (3 1) = 2 + 3 1 = 1.

2
1

1
1
1
2
1. 16a 2 y = 16 2 = 2 = 1 .
8
6
3
3
2. 2 + 0,3 (9) = 0,7 < 4,7 = 2 0,3 (9).
3. ) 5 + 7b 2a 8b = 3a b;
) 3 (4 + 2) 5 = 12 + 6 5 = 12 + 1;
) 20b (b 3) + (3b 10) = 20b b + 3 + 3b 10 = 22b 7.
4. 6 (0,5 1,5) 4,5 8 = 3 + 9 4,5 8 = 7,5 + 1;
2
7,5 x + 1 = 5 + 1 = 4 .
3
5. S = V1t + V2t; S .
S = 80 3 + 60 3 = 240 + 180 = 420 .
6. 2 (3 (2 )) = 2 3 + (2 ) = + 2 = .

1
1
1.

2
2
2 68
8
(1 17,6 : 55) = (1 0,32 ) =
=
= 0,08 .
17
17
17 100
100

2. 26 4 = 26 4 7,3 = 3,2.

140

3. ) 15 + 8 7 = 14 + ;
) 2 (5b 1) + 3 = 10b 2 + 3 = 10b + 1;
) 3 2 4 + 1 = 2 5;
) 4 (3b + 2) 2 (2b 3) = 12b + 8 4b + 6 = 8b + 14;
4.

2
(x 6 y ) 1 (2 x y ) = 2 x 4 y 2 x + 1 y = 1 y 4 y = 3 2 y .
3
3
3
3
3
3
3

5. S = Vt + ut = (V + u) t; S = (70 + 40) 2 = 220 /.

S .
6. 2 (3 (4 5)) = 2 3 + (4 5) = + 4 5 = 3 5.

2
1
1. (0,64 + 0,9) (65,7 69,2) = 1,54 (3,5) = 5,39.
2. 5

7
5 7 5 2
+ 2 = = .
15 6 3 3 3

3. ) 3a 7b 6a + 8b = 3a + b; ) 3 (4 + 2) 6 = 12 + 6 6 = 12;
) 10 (3 + 1) + ( 4) = 10 3 1 + 4 = 8 5;
) 2 (2 1) 3 ( + 2) = 4 2 3 6 = 8.
4. 0,5 (a 4b) + 0,1 (5a + 10b) = 0,5a 2b + 0,5a + b = a b.
5. 3 ;
= + 0,9 + (a + b) = 2,9a + b; x = 2,9 20 + 3 = 61 ().
6. 10 + (8 (6 + 4)) = 10 + 8 (6 + 4) = 18 6 4 = 12 4.

3
1
1. 4

3
1
1
3 = 3 = 3,5 .
4
6
2

2. 0,4 10 + 2 = 2 > 6 = 0,4 10 2.

3. ) 5 + 3 2 9 = 3 6; ) 2 (3 4) + 5 = 6 8 + 5 = 6 3;
) 15 ( + 3) + (2 1) = 15 3 + 2 1 = 16 4.
4. 2 (3,5 2,5) + 4,5 1 = 7 + 5 + 4,5 1 = 2,5 + 4;
25 4

+ 4 = 2 + 4 = 2 .
10 5
5. /.

141

: p = ut + xt; x =

p ut
9 12 0,5
= 6 /.
; x=
t
0,5

6. 5 (3 (2 4)) = 5 3 + (2 4) = 2 + 2 4 = 4 4.

4
1
1. 12

3
5
3 = 9 2,5 = 11,5 .
4
6

2. 1 0,6 5 = 2 < 1 + 0,6 5 = 4.

3. ) 12 10b 10a + 6b = 2a 4b; ) 4 (3 2) + 7=12 8 + 7=12 1;
) 8 (2 + 5) + ( 1) = 8 2 5 + 1 = 7 6.
4. 5 (0,6 1,2) 1,5 3 = 3 + 6 1,5 3 = 4,5 + 3;
45 4
+3 = 2+3 = 5.
10 9
5. S = Va + ua = (V + u) a; S = (5 + 4) 3 = 28 .

S .
6. 7 (5 (3 + )) = 7 5 + (3 + ) = 2 + 3 + = 5 + .

3
1
1.

4
(8,37 : 2,7 8,7 ) = 4 (3,1 8,7 ) = 4 56 = 3,2 .
7
7
7 10

2. 8 3,7 = 8 (2,6) 3,7 = 24,5.

3. ) 4b + 2y 12b y = 8b + y; ) 40 + 6 ( 7) = 40 + 6 42 = 6 2;
) 2 + (3 4) (4 7) = 2 + 3 4 4 + 7 = + 3;
) 3 ( 1) 2 (3 5) = 3 3 6 + 10 = 3 + 7.
4.

5
(12c + a ) + 2 (3c 2a ) = 10c + 5 a + 2c 4 a = 12c 1 a .
6
3
6
3
2

5. u t , V t;
. , S:
S = ut Vt = (u v) t; S = (60 10) 0,5 = 25 .
6. 10 (12 ( 6)) = 10 12 + ( 6) = 2 + 6 = 6.

142

 4
1
1. 2 6,72 : (15,42 13,02) = 2 6,72 : 2,4 = 2 2,8 = 0,8.
2. 4

7
5 7
1
+ 7 = +5 = 7 .
12
7 3
3

3. ) 14 14b + 10b 4c = 10c 4b;

) 5 (2 3) + 10 = 10 15 + 10 = 10 5;
) 5 + ( 4) (2 3) = 5 + 4 2 + 3 = 4 1;
) 4 ( 5) 2 ( + 3) = 4 20 2 6 = 2 26.
4. 0,4 ( 5) + 1,5 (2 ) = 0,4 2 + 3 1,5 = 3,4 3,5.
5. S , . :
S = a + (a b) + 0,8a = a + a b + 0,8a = 2,8a b;
S = 2,8 30 5 = 84 5 = 79 .
6. (3 (5 1)) = 3 + (5 1) = 2 + 5 1 = 3 1.

1.

2.
3.

4.

2
1
) 5 4,5 = 3 + 2,5;
) x = 12 ;
3
= 36;
2 = 7; = 3,5;
) 6 10,2 = 0;
) 2 (6 5) = 45;
6 = 10,2;
2 6 + 5 = 45;
= 1,7;
4 = 40; = 10.
; + 6 ;
+ + 6 = 26; 2 = 20; = 10 .
; 3 ;
3 20 = + 10; 2 = 304 = 15 2- .
: 45 + 15 = 60 ().
7 ( + 3) = 3 (2 1); 4 7 3 = 6 3;
0 = 0 , .

2
2
1
1. ) x = 18 ;
6
= 108;
) 7 + 11,9 = 0;

) 6 0,8 = 3 + 2,2;
3 = 3; = 1;
) 5 (7 + 7) = 9;
143

7 = 11,9;
= 1,7;

5 7 7 = 9;
2 = 16; = 8.

2. ; 9
+ 9 = 600; 10 = 600; = 60 .
3. . 5 .
5 50 = + 90; 4 = 140; = 35 ;
5 35 = 175 1- .
: 35 + 175 = 210 .
4. 6 (2 5) = 2 (2 + 4); 6 2 + 5 = 4 + 8;
0 = 3 , .

1
2
1. ) 3 + 2,7 = 0; 3 = 2,7; = 0,9;
) 2 + 7 = 3 2 (3 1); 2 + 7 = 3 6 + 2; 5 = 5; = 1;
2x x 3
=
; 4 = 5 15; = 15.
)
5
2
2. VII 4 4 VII ; + 2 VII ;
+ ( 4) + ( + 2) = 103; + 4 + + 2 = 103; 3 = 105;
= 35 VII ; 35 4 = 31 VII ;
35 + 2 = 37 VII .
3.

2x 1 x + 5 1 x
=

; 8 (2 1) = + 5 12 (1 );
3
8
2
1
16 8 = + 5 12 + 12; 3 = 1; x = .
3

4. ;
: 10 2- ;
4
(x + x 10) = 1,6 x 8 3- ;
5
+ 10 + 1,6 8 = 90; 3,6 = 108;
= 30 () ;
30 10 = 20 () ;
1,6 30 8 = 40 () .

2
2
1. ) 5 0,8 = 2 + 1; 3 = 2,4; = 0,8;

144

) 4 2 ( + 3) = 4 ( 5); 4 2 6 = 4 20; 6 = 18; = 3;

x 2x + 6
; 3 = 2 + 6; 3 = 3; = 1.
) 1 =
3
3
2. ;
+ 5 ;
6 = 4 ( + 5); 6 = 4 + 20; 2 = 20;
= 10 .
3.

1 2x x + 3 2 4x
;

=
3
4
5
20 40 15 45 = 24 48; 7 = 49; = 7.

4. ; 2 ;
2 5 = 3 ( 10); 2 5 = 3 30;
= 25 () ;
2 25 = 50 () ;
: 25 + 50 = 75 ().

3
2

1
x = 5 ; = 25;
5
) 3 11,4 = 0; 3 = 11,4; = 3,8;
) 4 + 5,5 = 2 2,5; 2 = 8; = 4;
) 2 (6 + 1) = 9; 2 6 1 = 9; 4 = 10; = 2,5.

1. )

2. ; + 7 ;
+ + 7 = 35; 2 = 28; = 14 .
3. ; 3 ;
3 30 = + 10; 2 = 40;
= 20 () ; 3 20 = 60 () ;
: 20 + 60 = 80 ().
4. 8 (2 + 4) = 2 (3 2); 8 2 4 = 6 4;
0 = 0 , .

4
2

1
x = 8 ; = 32;
4
) 5 12,5 = 0; 5 = 12,5; = 1,5;

1. )

145

) 3 0,6 = + 4,4; 2 = 5; = 2,5;

) 4 (7 2) = 17; 4 7 + 2 = 17; 3 = 15; = 5.
2. . = 4 ;
4 + = 60; 5 = 60; = 12 ().
3. ;
5 ;
5 25 = + 15; 4 = 40; = 10 ();
: 10 + 5 10 = 10 + 50 = 60 ().
4. 3 (9 3) = 3 (4 2); 3 9 + 3 = 12 6;
0 = 9 , , .

3
2
1. ) 4 6,4 = 0; 4 = 6,4; = 1,6;
) 5 + 3 = 7 5 (2 + 1); 5 + 3 = 7 10 5; 8 = 8; = 1;
x4
3x
2 =
; 4 10 = 3; 2 = 14; = 7.
)
5
5
2. 2- ; 2 ;
+ 10 ; + 2 + + 10 = 70; 4 = 60;
= 15 () 2- ;
2 15 = 30 () 1- ;
15 + 10 = 25 () 3- .
3.

x 1 2x x + 3
;

=
2
3
5
15 ( 1) 10 2 = 6 ( + 3); 15 15 20 = 6 + 18;
11 = 33; = 3.

4. 2 ; 2 1 ;
1,5 (2 30) = + 5; 3 45 = + 5; 2 = 50; = 25 ();
: 25 + 2 25 = 25 + 50 = 75 ().

4
2
1. ) 7 2 = 4,5 7; 5 = 2,5; = 0,5;
) 2 ( 8) 5 ( + 6) = 2; 2 16 5 30 = 2; 3 = 48; = 16;
6x x 5
=
; 2 6 = 7 ( 5); 12 = 7 35; 5 = 35; = 7.
)
7
2
2. / ;

146

 + 2 / ; 2 / ;
4 ( + 2) = 5 ( 2); 4 + 8 = 5 10; = 18 /.
3.

3x 7 2 x x + 4
=

; 3 7 = 4 3 ( + 4);
6
3
2
3 7 = 4 3 12; 2 = 5; = 2,5.

4. 1- ;
+ 1 ;
2
(x + x + 1) = 4 x + 2 3- ;
3
3
3
4
2
10
40
x + x + 1 + x + = 15 ;
x=
;
3
3
3
3
= 4 () 1- ; 4 + 1 = 5 () 2- ;
4
2
4 + = 6 () 3- .
3
3

1
3
1. ) = 6 0,5 + 19 = 22;
) 6 + 19 = 1; 6 = 18; = 3;
) 7=6 (2) + 19; 7=7 , (2; 7);
2. )

3.

= -2

= 2 4

=3

3
2
1

1
0

-1 0

-4

) = 1.
y = 47 x 37
4.
;
y = 13x + 23

y = 47 x 37
;

47 x 37 = 13 x + 23
147

 y = 10
; (1; 10).

x = 1
5. y = kx + b; k = 3; y = 3x + b;
y = 3x + b (0; 0) , ;
0 = 3 0 + b; b = 0.
: = 3.

2
3
1. ) = 40;
) 4 30 = 6; 4 = 24; = 6;
) 3 = 4 7 30;
3 = 2 , (7; 3);
2. )
= -3 + 3

3.

= 0,5

3
1
0

1
1

= -4
-4

) = 1.
y = 38 x + 15
4.
;
y = 21x 36
y = 38 x + 15
;

17 x = 51

y = 38 x + 15
;

38 x + 15 = 21x 36
y = 99
(3; 99).

x = 3

5. y = kx + b; k = 5; y = 5x + b; y = 5x + b (0; 0) ,
0 = 5 0 + b; b = 0. : = 5.

1
3
1. ) (0,6 53 15) 2 = 602 = 3600; ) 0,3 (18) 2 = 97,2.

148

2. ) 10 15 = 10+15 = 25; ) 16 : 11 = 1611 = 5;

4

a4
a
) (7) 3 = 73 = 21; ) () 6 = 66; ) =
.
625
5
3. 27000 = 2,7 104.
4. ) 4a7b5 (2ab2) = 8a8b7; ) (342) 3 = 27126; ) (25) 2 = 4102.
5. ) 79 711 : 718 = 79+1118 = 72 = 49;
) 56 125 : 254 = 56 53 : (52) 4 = 56+324 = 51 = 5.
2

2 5 8 1 3 6
8
9
x y 1 x y = x 5 y 8 x 6 y 12 = 6 x 11 y 20 ;
3
3
4

2
) (a1+n) 3 : a3 = a3 (n+1) 3 = a3n.

6. ) 2

2
3
3

1. ) 16 6 2 = (4 )3 = 64 ; ) 5 0,82 = 3,2.
3

2. ) 12 10 = 12+10 = 22; ) 18 : 13 = 1813 = 5.

3. 3800 = 3,8 103.
4. ) 35 4ab6 = 12a6b6; ) (26) 4 = 16424; ) (3a3b4) 3 = 27a9b12.
5. ) 615 611 : 624 = 615+1124 = 62 = 36;
) 311 27 : 96 = 311 33 : (32) 6 = 311+362 = 32 = 9.
4

1
16
1
1
1

6. ) 3 a 8 b a 3 b 8 = q 8 b a 12 b 32 = a 20 b 33 ;
5
5
16
5
2

) xn2 x2 xn+2 = xn2+2+n+2 = x2n+2.

3
3
1. ) = 20; ) 3 = 5 + 18; 5 = 15; = 3;
) 12 = 5 (6) + 18;
12 = 12 , (6; 12).
2. )

3.

149

= 2 + 4

=5

4
= -0,5
1

1
-2

-2

) = 1.
y = 14 x + 32
4.
;
y = 26 x 8

y = 14 x + 32
;

26 8 x = 14 x + 32

y = 18
; (1; 18).

x = 1

5. y = kx + b; k = 2; y = 2x + b;
y = 2x + b (0; 0) , 0 = 2 0 + b; b = 0.
: = 2.

4
3
1. ) = 22;
) 5 = 2 15; 2 = 10; = 5;
) 5 = 2 10 15;
5 = 5 , , K (10; 5).
2. ) 15 3 = 15+3 = 18; ) 20 : 12 = 2012 = 8; ) (3) 6 = 36 = 18;
2

y2
y
.
) () 5 = 55; ) =
25
5
3. 56000 = 5,6 104;
4. ) 26 (427) = 888; ) (5b2) 5 = a25b10; ) (24) 2 = 428.
5. ) 824 : 816 : 86 = 824166 = 82 = 64;
) 25 8 : 43 = 25 23 : (22) 3 = 25+323 = 22 = 4.
2

2
25 2 10 27 3 10
2

6. ) 1 xy 5 5 x 3 y 10 =
x y
x y = 15 x 5 y 20 ;
5
9
5
3

) (cn+1) 2 cn = c2n+2+n = c3n+2.

150

 4
3
3

1. ) 40 12 2 = 4 3 = 64 ; ) 0,4 53 = 50.
4

2. ) 4 11 = 4+11 = 15; ) 16 : 10 = 1610 = 6; ) (5) 3 = 53 = 15;

3

p3
p
.
) () 4 = 44; ) =
27
3
3. 2100 = 2,1 103.
4. ) 62 (35) = 1826; ) (4a3b) 2 = 16a6b2; ) (74) 3 = 2112.
5. ) 511 52 : 510 = 511+210 = 53 = 125;
) 166 : 47 : 64 = (42) 6 : 47 : 43 = 41273 = 42 = 16.

1
7 3 12 24 9 5 12
a b
a b = 16 a 17 b 36 ;
3
27
7
) a2n+5 : (an) 2 = a2n+52n = a5.

6. )

1
4
1. 1 5 (4) 2 = 1 80 = 79.
2. ) 7 12 = 7+12 = 19;
) 20 : 5 = 205 = 15;

) (2) 8 = 28 = 16;
) (2) 4 = 244 = 164.

3. ) 2ab3 3a2 b4 = 6a3b7; ) (2a5b2) 3 = 8a15b6.

4. = 1,5, = 2,25; = 1,5, = 2,25.

= 2

1
0

151

5. (52) 2 55 : 57 = 522+57 = 52 = 25.

4

2
8
81
1
1

6. ) 2 x 2 y8 1 xy 3 = x 2 y8 x 4 y12 = 13 x 6 y 20 ;
3
2
3
16
2

n2
3n
n2+3n+1
2
) x x = x
=x.

2
4
3

9 1
1
1. 9 =
= .
27 3
3
2. ) 3 22 = 3+22 = 25;
) 18 : 6 = 186 = 12;
) (4) 6 = 46 = 24;
) (3) 5 = 355 = 2435.
3. ) 452 34 = 1266;
) (323) 2 = 946.
4. = 4 = 2 = 2.

= 2

1
0

5. 36 33 : (34) 2 = 36+342 = 31 = 3.
2

3
1
24 5 6 49 10 2
2

x y x y = 18 x15 y8 ;
6. ) 3 x5 y 6 2 x5 y =
7
3
7
9
3

) (an+1) 2 : a2n = a2 (n+1) 2n = a2.

152

 1
4
1. ) (3ab + 5a b) (12ab 3a) = 3ab + 5a b 12ab + 3a =
= 8a 9ab b;
) 22 (3 53) = 62 105;
) (2 3) ( + 2) = 22 + 4 3 62 = 22 + 62;
) ( 1) (2 + 2 4) = 3 + 22 4 2 2 + 4 = 3 + 2 6 + 4;
) (33 62) : 32 = ( 2) 32 : 32 = 2.
2. 3 ( 2) ( 3) ( 1) = 32 6 2 + + 3 3 = 22 2 3.
3. 0,3 (42 3) (22 + 5) = 0,3 (84 + 202 62 15) = 0,3 (84 +
+ 142 15) = 2,45 4,23 + 4,5.
4. 2 (a + b c) 2b (a b c) + 2c (a b + c) = 2a2 + 2ab 2ac
2ab + 2b2 + 2bc + 2ac 2bc + 2c2 = 2a2 + 2b2 + 2c2.
5. ;
( + 2) ( + 3) ;
: ( + 2) ( + 3) 2 = 51;
2 + 5 + 6 2 = 51;
5 = 45;
= 9 ().

2
4
1. ) 152 + 7 (13 52) = 152 + 7 13 + 52 = 202 6;
) 2 ( 3b + 4) = 2ac 6bc + 8c;
) (4 1) (2 3) = 82 12 2 + 3 = 82 14 + 3;
) ( + 2) (2 3) = 3 2 3 + 22 2 6 = 3 + 2 5 6;
) (4ab2 6a2b) : 2ab = (2b 3a) 2ab : 2ab =2b 3a.
2. 2 (3 4) 3 (3 1) = 62 8 92 + 3 = 32 5.
3. 1,5 (32 5) (22 + 3) = 1,5 (64 + 92 102 15) = 1,5 (64 2
15) = 95 1,53 22,5.
4. 5 ( + b + c) 5b (a b c) 5c (a + b c) = 5a2 + 5ab + 5ac
5ab + 5b2 + 5bc 5ac 5bc + 5c2 = 5a2 + 5b2 + 5c2.
5. ; + 1 ;
.. , , 1 , : ( 2) ( + 1 2) . : ( + 1) ( 2) ( 1) = 22;
2 + 2 + 3 2 = 22; 4 = 24;
= 6 () ; 6 + 1 = 7 () .

153

 3
4
1. 3 (5) 4 + 7 = 7 75 = 68.
2. ) 8 16 = 8+16 = 24;
) 16 : 4 = 164 = 12;

) (3) 5 = 35 = 15;
) (2) 3 = 233 = 83.

3. ) 3a2b (2a3b4) = 6a5b5;

) (3a3b2) 3 = 27a9b6.
4. = 9 = 3 = 3.

= 2

1
0

5. 56 53 : (52) 4 = 56+324 = 51 = 5.
4

1
1
54 12 4 16 8 5

6. ) 2 a 3b 3 a8b5 =
a b a b = 125a 20b9 ;
2
5
16
5

2n
n1 2
2n2 (n1)
2n2n+2
2
=x
=x.
) x : (x ) = x

3
4
1. ) (2 3 + 7) (3 5) = 2 3 + 7 3 + 5 = 2 + 7;
) 33 (22 4) = 65 123;
) (2 + ) (3 ) = 62 2 + 3 2 = 62 + 2;
) ( + 1) (3 3 4) = 3 32 4 + 2 3 4 = 3 22 7 4;
) (84 + 23) : 23 = (4 + 1) 23 : 23 = 4 + 1.
2. ( 4) ( 5) 2 ( 6) = 2 5 4 + 20 22 + 12 = 2 + 3 + 20.
3. 0,5 (4 22) (2 + 3) = 0,5 (42 + 12 24 62) = 0,5 (24
22 + 12) = 5 + 3 6.
4. 7 ( + ) 7 ( ) + 7 ( + ) = 7 + 72 7 7 +
+ 7 + 72 + 72 7 + 7 = 72 + 72 + 72.

154

5. .
( 1) ( 3) ;
: 2 ( 1) ( 3) = 21; 2 2 + 4 3 = 11; 4 = 24; = 6 ().

4
4
1. ) (12 62 + 5) (2 32) = 12 62 + 5 2 + 32 = 32 +
+ 10 + 5;
) 2 (2 + 3 1) = 23 + 62 2;
) (3 5) (4 3) = 122 9 20 + 15 = 122 29 + 15;
) ( + 4) (2 + 2 3) = 3 + 22 3 + 42 + 8 12 = 3 + 62 + 5 12;
) (152 + 10) : 5 = (3 + 2) 5 : 5 = 3 + 2.
2. 3 (2 + 4) 2 (2 3) = 62 + 12 42 + 6 = 22 + 18.
3. 0,4b(5b2 10) (2 + b2)=0,4b(10b2 + 5b4 20 10b2)=0,4b (5b4 20)=
= 2b5 8b.
4. 3c (a + b c) + 3b (a b c) 3a (a + b + c) = 3ac + 3bc 3c2 + 3ab
3b2 3bc 3a2 3ab 3ac = 3a2 3b2 3c2.
5. . + 15 . , (
+ 2) ( + 15 + 2) .
: ( + 2) ( + 17) ( + 15) = 74;
2 + 19 + 34 2 15 = 74; 4 = 40;
= 10 () ; 10 + 15 = 25 () .

1
5
1. ) (3 4 + 2) (11 14) = 3 4 + 2 11 + 14 =
= 10 8 + 2;
) 32 (3 + 1) = 35 + 32.
2. ) 10ab 15b2 = 5b (2a 3b); ) 183 + 62 = 62 (3 + 1).
3. 9 6 ( 1) 5 ( + 2) = 9 6 + 6 = 5 + 10;
2 = 4; = 2.
4. / .
20 / .
4 = 6 ( 20); 4 = 6 120; 2 = 120;
= 60 (/) .
5.

3x 1 x 5 x
;
=
6
3
9

155

3 (3 1) 6 = (5 ) 2; 9 3 6 = 10 2;
5 = 13; = 2,6.
6. 2 (a + b + c) 2b (a b c) + 2c (a b + c) = 2a2 + 2ab 2ac
2ab + 2b2 + 2bc + 2ac 2bc + 2c2 = 2a2 + 2b2 + 2c2.

2
5
1. ) (22 3 + 1) (72 5) = 22 3 + 1 72 + 5 = 52 + 2 + 1
) 3 (42 ) = 123 32.
2. ) 2 32 = (2 3);
) 8b4 + 2b3 = 2b3 (4b + 1).
3. 7 4 (3 2) = 5 (1 2);
7 12 + 4 = 5 10; 2 = 6; = 3.
4. VI .
( + 2) VI , ( + 2 + 3) VI ;
+ + 2 + + 5 = 91; 3 = 84;
= 28 () VI A; 28 + 2 = 30 () VI ;
28 + 5 = 33 () VI .
5.

x 1 5 x 3x
=
+
;
5
2
4
4 ( 1) = 10 (5 ) + 15; 4 4 = 50 10 + 15; = 54;

6. 3 (x + y + ) 3y (x y c) 3 ( + ) = 32 + 3 + 3 3 +
+ 32 + 3 3 3 + 32 = 32 + 32 + 32.

1
5
1. ) 22 = (2 ); ) ab + 3ab2 = ab (1 + 3b);
) 24 + 63 42 = 22 (2 + 3 2);
) 2 ( 1) + 3 ( 1) = ( 1) (2 + 3);
) 4 4 + = 4 ( ) + ( ) = ( ) ( + 4).
2. ) 2a2b2 6ab3 + 2a3b = 2ab (ab 3b2 + a2);
) 2 ( 2) ( 2) 2 = ( 2) ( ( 2)) = 2 ( 2);
) 3 3 + 2 = 3 ( ) ( ) = ( ) (3 );
) + + = ( ) ( ) + =
= ( ) ( + 1).
3. 2 2 + 2 = ( ) 2 ( ) = ( ) ( 2);

156

2 2
2 2
2

3 2 2 2 = 1 = .
3 3
3 3
3

2
5
1. ) 62 + ab 5a = a (6a + b 5); ) 72 2 = (7 );
) 125 + 43 = 43 (3 + 1);
) 3 ( + 2) 2 ( + 2) = ( + 2) (3 2);
) ab + 2ac + 2b + 4c = a (b + 2c) + 2 (b + 2c) = (b + 2c) (a + 2).
2. ) 33 + 622 332 = 32 ( + 2 );
) 2 (1 ) + ( 1) 2 = (1 ) ( + 1 ) = (1 );
) 2a + ab 2b b2 = a (2 + b) b (2 + b) = (b + 2) (a b);
) 5a 5b xa + xb b + a = 5 (a b) x (a b) + a b =
= (a b) (5 x + 1) = (a b) (6 x).
3. 4 4 + 2 = 4 ( ) ( ) = ( ) (4 );
(3,5 + 1,5) (4 3,5) = 5 0,5 = 2,5.

3
5
1. ) (12ab 5a) (ab + 6a) = 12ab 5a ab 6a = 11ab 11a;
) 5 (32 2 4) = 153 102 20.
2. ) 32 + 9 = 3 ( + 3); ) 105 5 = 5 (24 1).
3. 4 ( + 1) = 15 7 (2 + 5);
4 + 4 = 15 14 35;
3 = 39;
= 13.
4. ;
+ 6 .
8 = 5 ( + 6); 3 = 30;
= 10 .
5.

2 x 2 x + 1 3x 5
;

=
3
6
4
8 2 (2 + 1) = 3 (3 5); 8 4 2 = 9 15;
5 = 13; = 2,6.

6. 4 ( + + ) + 4 ( ) 4 ( ) = 4 + 42 + 4 + 42
4 4 4 + 4 + 42 = 42 + 42 + 42.

157

 4
5
1. ) (43 + 15) (17 3) = 43 + 15 17 + 3 = 53 2;
) 2 (3a b + 4) = 6a2 2ab + 8a.
2. ) 2ab ab2 = ab (2 b); ) 22 + 46 = 22 (1 + 24).
3. 5 ( 3) = 14 2 (7 2); 5 15 = 14 14 + 4; = 15.
4. ; + 12 ;
2 ;
+ + 12 + 2 = 56; 4 = 44;
= 11 () ;
11 + 12 = 23 () ;
2 11 = 22 () .
5.

3 x x + 1 5x
;
=

3
2
4
4 (3 ) = 6 ( + 1) 15;
12 4 = 6 + 6 15;
5 = 6; = 1,2.

6. 6 ( + ) + 6 ( + ) 6 ( ) = 62 6 + 6 + 6 +
+ 62 6 6 + 6 + 62 = 62 + 62 + 62.

3
5
1. ) 3ab + a2 = a (3b + a);
) 2 42 = 2 (1 2);
) 4b3 3b5 = b3 (4 3b2);
) 5 ( + ) + ( + ) = ( + ) (5 + );
) 2 + 2b bx = a (2 x) + b (2 x) = (2 x) (a + b).
2. ) 322 63 + 122 = 32 ( 2 + 4);
) ( + 3) 2 2 ( + 3) = ( + 3) ( + 3 ) = 3 ( + 3);
) 4 + 2 22 = 2 (2 + 1) (2 + 1) = (2 + 1) (2 );
) 3 3 + + = 3 ( ) + ( ) + = ( ) (3 +
+ + 1) = ( ) ( + 4).
3. 2 2 + 2 = 2 ( ) ( ) = ( ) (2 );
1
1
1
1

2 + 3 2 + 2 = 4 .
7
7
7
7

158

 4
5
1. ) 3 2 = (3 );
) ab2 + 5a2b = ab (b + 5a);
) 24 43 + 2 = 2 (3 22 + 1) = 2 (3 2 + 1 2) = 2 (2 (
1) ( 1) ( + 1)) = 2 ( 1) (2 1);
) b (b 3) 4 (b 3) = (b 3) (b 4);
) 2 + 2 + ab + b = 2c (a + 1) + b (a + 1) = (a + 1) (2c + b).
2. ) 12a2b2 + 6a2b3 + 12ab3 = 6ab2 (2a + ab + 2b);
) b (b 2) 2 + b2 (2 b) = b (b 2) (b 2 b) = 2b (b 2) = 2b (2 b);
) 5 2 + 5 = ( 5) ( 5) = ( 5) ( );
) ab ac + 2c 2b b + c = a (b c) 2 (b c) (b c) =
= (b c) (a 2 1) = (b c) (a 3).
3. bc + b2 3c 3b = b (c + b) 3 (c + b) = (b + c) (b 3);
(3,7 4,7) (3,7 3) = 0,7.

1
6
1. ) ( + 2) ( 3) = 2 3 + 2 6 = 2 6;
) (2 1) (3 + 4) = 62 + 8 3 4 = 62 + 5 4;
) (5 2) (4 ) = 202 5 8 + 22 = 202 13 + 22;
) ( 2) (2 3 + 6) = 3 32 + 6 22 + 6 12 = 3 52 +
+ 12 12.
2. ) ( + 3) 2 ( + 3) = ( + 3) ( 2);
) + 5 5 = ( ) + 5 ( ) = ( ) ( + 5).
3. 0,1 (22 + 6) (5 42) = 0,1 (102 84 + 30 242) = 0,1 (84
142 + 30) = 0,85 + 1,43 3.
4. ) 2 4 + 4 = ( ) 4 ( ) = ( ) ( 4);
) ab ac bx + cx + c b = a (b c) x (b c) (b c) =
= (b c) (a x 1).
5. ;
( + 2) ( + 3) ;
( + 2) ( + 3) 2 = 51; 2 + 5 + 6 2 = 51; 5 = 45;
= 9 ().

159

 2
6
1. ) ( 5) ( 3) = 2 3 5 + 15 = 2 8 + 15;
) (5 + 4) (2 1) = 102 5 + 8 4 = 102 + 3 4;
) (3 + 2) (2 + 4) = 62 + 12 + 4 + 82 = 62 + 16 + 82;
) (b 2) (b2 + 2b 3) = b3 + 2b2 3b 2b2 4b + 6 = b3 7b + 6.
2. ) ( ) + ( ) = ( ) ( + );
) 2 2b + ca cb = 2 (a b) + c (a b) = (a b) (c + 2).
3. 0,5 (42 1) (52 + 2) = 0,5 (204 + 82 52 2) = 0,5 (204 + 32
2) = 105 + 1,53 .
4. ) 2 2 + 2 = (2 ) (2 ) = (2 ) ( );
) bx + by x y ax ay = b (x + y) (x + y) a (x + y) =
= (x + y) (b 1 a).
5. ; + 6 ;
( + 1) ( + 6 + 1) .
: ( + 1) ( + 7) ( + 6) = 15;
2 + 8 + 7 2 6 = 15; 2 = 8;
= 4 () ; 4 + 6 = 10 () .

1
6
1. ) ( 3) 2 = 2 6 + 9; ) (2 + ) 2 = 42 + 4 + 2;
2. ) 4 ( 2) ( 4) 2 = 42 8 2 + 8 16 = 32 16;
) 2 (b + 1) 2 4b = 2b2 + 4b + 2 4b = 2b2 + 2.
3. ) 2 25 = ( 5) ( + 5);
) ab2 ac2 = a (b2 c2) = a (b c) (b + c);
) 32 6ab 3b2 = 3 (a2 + 2ab + b2) = 3 (a + b) 2.
4. (2 2) 2 2 ( + 3) ( 3) + 2 (22 + 5) = 4 43 + 42 4 + 92 +
+ 43 + 10 = 132 + 10.
5. ) 252 ( + 3) 2 = (5 3) (5 + + 3) = (4 3) (6 + 3) =
= 3 (2 + 1) (4 3);
) 27a3 + b3 = (3a + b) (9a2 3ab + b2);
) 164 81 = (42 9) (42 + 9) = (2 3) (2 + 3) (42 + 9);
) 2 2 = ( ) ( + ) ( + ) = ( + ) ( 1).

160

 2
6
1. ) ( + 4) 2 = 2 + 8 + 16;
) ( 2b) 2 = a2 4ab + 4b2;
) (3 + 5) (3 5) = 92 25.
2. ) ( 2) ( + 3) ( 1) 2 = 2 + 3 2 6 2 + 2 1 = 3 7;
) 3 ( + ) 2 6 = 32 + 6 + 32 6 = 32 + 32.
3. ) 162 9 = (4 3) (4 + 3);
) 33 75 = 3 (2 25) = 3 ( 5) ( + 5);
) 22 + 4 + 22 = 2 (2 + 2 + 2) = 2 ( + ) 2.
4. (6 2) 2 2 ( 1) ( + 1) + 6 (3 + 22) = 362 123 + 4 4 + 2 +
+ 18 + 123 = 372 + 18.
5. ) ( + 2) 2 42 = ( + 2 2) ( + 2 + 2) = (2 ) (3 + 2);
) 3 83 = ( 2) (2 + 2 + 42);
1
1
1

) 16 x 4 = 4 x 2 4 + x 2 = ;
9
9
81

1
1
1

= 2 x 2 + x 4 + x 2 ;
3
3
9

) 2 + 2 + 2 2 = 2 ( + ) + ( ) ( + ) = ( + ) (2 + ).

3
6
1. ) ( 8) ( + 5) = 2 + 5 8 40 = 2 3 40;
) (3b 2) (4b 2) = 12b2 6b 8b + 4 = 12b2 14b + 4;
) (6 + ) (2 3) = 122 18 + 2 32 = 122 16 32;
) ( + 1) (2 + 3 + 2) = 3 + 32 + 2 + 2 + 3 + 2 = 3 + 42 + 5 + 2.
2. ) 2 ( 1) 3 ( 1) = ( 1) (2 3);
) ab + ac + 4b + 4c = a (b + c) + 4 (b + c) = (b + c) (a + 4).
3. 0,4 (22 + 3) (5 32) = 0,4 (102 64 + 15 92) = 0,4 (64 +
+ 2 + 15) = 2,45 0,43 6.
4. ) a2 + ab 3a 3b = a (a + b) 3 (a + b) = (a + b) (a 3);
) kp kc px + cx + c p = k (p c) x (p c) (p c) =
= (p c) (k x 1).
5. ;
( 2) ( 3) .
: 2 ( 2) ( 3) = 24; 2 2 + 5 6 = 24;
5 = 30;

161

 = 6 () .

4
6
1. ) ( 4) ( 2) = 2 2 4 + 8 = 2 6 + 8;
) (3 + 1) (5 6) = 152 18 + 5 6 = 152 13 6;
) (3 2) ( + 6) = 32 + 18 2 122 = 32 + 16 122;
) (b + 3) (b2 + 2b 2) = b3 + 2b2 2b + 3b2 + 6b 6 = b3 + 5b2 +
+ 4b 6.
2. ) 2 (a b) + a (a b) = (a b) (2x + a);
) 3 + 3 + bx + by = 3 (x + y) + b (x + y) = (x + y) (b + 3).
3. 0,2 (52 1) (22 + 1) = 0,2 (104 + 52 22 1) = 0,2 (104 + 32
1) = 25 + 0,63 0,2.
4. ) 3 3 + 2 = (3 ) (3 ) = (3 ) ( );
) + + = ( ) ( ) ( ) =
= ( ) ( 1).
5. ;
( + 5) ;
, ( + 2) ( + 5 + 2) ;
: ( + 2) ( + 7) ( + 5) = 26;
2 + 9 + 14 2 5 = 26; 4 = 12;
= 3 () ; 3 + 5 = 8 () .

3
6
1. ) (2 1) 2 = 42 4 + 1;
) ( + 3) 2 = 2 + 6 + 92;
) (7 ) (7 + ) = 49 2.
2. ) ( + 5) 2 5 (2 ) = 2 + 10 + 25 10 + 52 = 62 + 25;
) 16 + 2 ( 4) 2 = 16 + 22 16 + 32 = 22 + 32.
3. ) 81 2 = (9 ) (9 + );
) 52 52 = 5 (2 2) = 5 ( ) ( + );
) 32 6 + 32 = 3 (2 2 + 2) = 3 ( ) 2.
4. (2 + 4) 2 2 ( 2) ( + 2) 42 (2 1) = 4 + 83 + 162 4 +
+ 42 83 + 42 = 242.
5. ) ( 2) 2 16 = ( 2 4) ( 2 + 4) = ( 6) ( + 2);

162

1 3
1
1
1

y = a + y a 2 ay + y 2 ;
8
2
2
4

) 814 4 = (92 2) (92 + 2) = (3 ) (3 + ) (92 + 2);

) 2 + b + b2 = (a + b) + (b a) (b + a) = (a + b) (1 + b a).
) a 3 +

4
6
1. ) (2 + 3) 2 = 4 + 12 + 92;
) ( 5b) 2 = a2 10ab + 25b2;
) ( + 10) ( 10) = 2 100.
2. ) ( 4) 2 ( + 1) ( + 2) = 2 8 + 16 2 3 2 = 11 + 14;
) 5 (a + b) 2 10ab = 5a2 + 10ab + 5b2 10ab = 5a2 + 5b2.
3. ) 92 25 = (3 5) (3 + 5);
) 4 3 = (4 2) = (2 ) (2 + );
) 22 + 4 22 = 2 (2 2 + 2) = 2 ( ) 2.
4. (2b + b2) 2 + b2 (5 b) (5 + b) 4b (b2 3) = 4b2 + 4b3 + b4 + 25b2
b4 4b3 + 12b = 29b2 + 12b.
5. ) 16 ( + 1) 2 = (4 1) (4 + + 1) = (3 ) ( + 5);
1 3
1
1
1

)
a x 3 = a x a 2 + ax + x 2 ;
27
3
3
9

) 4 16b4 = (a2 4b2) (a2 + 4b2) = (a 2b) (a + 2b) (a2 + 4b2);

) 3 2 3 + 2 = ( ) ( + ) 3 ( ) = ( ) ( + 3).

1
7
1. ) ( 4) 2 = 2 8 + 16;
) (7 + ) 2 = 492 + 14 + 2;
) (5 1) (5 + 1) = 252 1;
) (3 + 2b) (3a 2b) = 9a2 4b2.
2. ( 9) 2 (81 + 2) = 2 18 + 81 81 2 = 2 20.
3. ) 2 49 = ( 7) ( + 7);
) 252 10 + 2 = (5 ) 2 = (5 ) (5 ).
4. (2 ) 2 ( + 1,5) = 4;
4 4 + 2 2 1,5 = 4;
5,5 = 0;
= 0.

163

5. ) (2 2) (2 + 2) = 4 42;
) (32 + ) 2 = 94 + 63 + 2;
) (2 + m) 2 (2 m) 2 = ( (2 + m) (2 m)) 2 = (4 m2) 2 = 16 8m2 + m4.
6. ) 422 94 = (2 32) (2 + 32);
) 252 ( + 3) 2 = (5 3) (5 + + 3) = (4 3) (6 + 3) =
= 3 (2 + 1) (4 3);
) 27m3 + n3 = (3m + n) (9m2 3mn + n2).

2
7
1. ) (3 + 4) 2 = 92 + 24 + 16;
) (2 b) 2 = 4x2 4bx + b2;

) (b + 3) (b 3) = b2 9;
) (5 2) (5 + 2) = 252 42.

2. (c + b) (c b) (5c2 b2) = c2 b2 5c2 + b2 = 4c2.

3. ) 252 2 = (5 ) (5 + );
) 2 + 4bc + 4b2 = (c + 2b) 2 = (c + 2b) (c + 2b).
4. 12 (4 ) 2 = (3 ); 12 16 + 8 2 = 3 2; 5 = 4; = 0,8.
5. ) (3 + 2) (3 2) = 92 4; ) (3 6) 2 = 6 124 + 362;
) ( ) 2 ( + ) 2 = ( ( ) ( + )) 2 = (2 2) 2 = 4 222 + 4.

1
1
1

6. ) 100a 4 b 2 = 10a 2 b 10a 2 + b ;

3
3
9

) 92 ( 1) 2 = (3 + 1) (3 + 1) = (2 + 1) (4 1);
) 3 + 6 = ( + 2) (2 2 + 4).

1
7

2b 4
x +1
; )
;
3a
x
a + 2b
1
=
.
)
(a 2b )(a + 2b ) a 2b

1. )

2 x 2 + 2ax 2ax + 2a 2 2 x 2 + 2a 2
= 2
;
(x a )(x + a )
x a2
2 ab + 2ab
2 + ab
=
;
)
a (b + 2 )
a (b + 2 )

2. )

164

)
3.
4.

c2 + c c2
c
.
=
c +1
c +1

7 x + 5 x + 5 y 12 x + 12 y

xx y

a (x y ) + 3(x y )
2

a 9
5,8 3,4
= 24 .
3,1 3

17 y

x x2 y2

).

(a + 3)(x y ) = x y , 3;
(a + 3)(a 3) a 3

2
7
1. )

3y 2
2x

; )

b(a 1)
b

(2 x y )(2 x + y ) = 2 x + y .
a 1
; )
b
2x y

2x 2 2x 2 + 4x
4x
3a + 15 + a 2 3a a 2 + 15
= 2
;
=
; )
a (a + 5)
a (a + 5 )
x2 4
x 4
7 a 7 a + 7b
7b
=
)
.
a b
a b

2. )

3.
4.

5a 10 5a 10 5a 2 + 20

5a 2

(a + 2 ) (a 2)
(a + 2 )2 (a 2 )
2(a c ) + x(a c ) (a c )(x + 2 ) a c
=
=
, 2;
(x 2 )(x + 2 ) x 2
x2 4
2

6,7 5,3
= 14 .
1,9 2

3
7
1. ) ( + 6) 2 = 2 + 12 + 36;
) (3 1) 2 = 92 6 + 1;

) (3 2) (3 + 2) = 92 4;
) (4 + 3k) (4a 3k) = 16a2 9k2.

2. (b 8) 2 (64 6b) = b2 16b + 64 64 + 6b = b2 10b.

3. ) 25 2 = (5 ) (5 + );
) a2 6ab + 9b2 = (a 3b) 2 = (a 3b) (a 3b).
4. 36 (6 ) 2 = (2,5 ); 36 36 + 12 2 = 2,5 2; 9,5 = 0; = 0.
5. ) (2 3) (3 + 2) = 4 92; ) (3 + 3) 2 = 92 + 64 + 6;
) (3 k) 2 (3 + k) 2 = ( (3 k) (3 + k)) 2 = (9 k2) 2 = 81 18k2 + k4.

165

6. ) 364 25a2b2 = (6a2 5ab) (6a2 + 5ab) = a2 (6a 5b) (6a + 5b);
) ( 7) 2 81 = ( 7 9) ( 7 + 9) = ( 16) ( + 2);
) a3 8b3 = (a 2b) (a2 + 2ab + 4b2).

4
7
1. ) (2 1) 2 = 42 4 + 1;
) (3 + ) 2 = 92 + 6 + 2;

) ( 5) ( + 5) = 2 25;
) (4b + 5c) (4b 5c) = 16b2 25c2.

2. ( + ) ( ) (2 + 32) = 2 2 2 32 = 42.
3. ) 162 0,25 = (4 0,5) (4 + 0,5);
) 2 + 10ab + 25b2 = (a + 5b) 2 = (a + 5b) (a + 5b).
4. (5 ) 2 (2,5 + ) = 0; 25 10 + 2 2,5 2 = 0; 12,5 = 25; = 2.
5. ) (2a b2) (2a + b2) = 4a2 b4;
) ( 63) 2 = 2 124 + 366;
) (y + b) 2 (y b) 2 = ( (y + b) (y b)) 2 = (y2 b2) 2 = y4 2b2y2 + b4.
1 2

1
1
a 0,09c 4 = a 0,3c 2 a + 0,3c 2 ;
81

9
9
) (b + 8) 2 4b2 = (b + 8 2b) (b + 8 + 2b) = (8 b) (8 + 3b);
) 9 b3 = (a3 b) (a6 + a3b + b2).

6. )

3
7
1. )

2. )

)
3.
4.

166

2x 6
3y

x(x y )

; )

x y
(3a 4)(3a + 4 ) = 3a 4 .
; )
3a + 4
x

yx
x y
1
2x 2 + 4x x 2 + 2x x 2 + 6x
=
= ;
= 2
; )
(x 2 )(x + 2)
x
(
x
y
)
x
(
x
y
)
x

x 4
3b 2 + 9b 9b 3b 2
=
.
b+3
b+3

3ab 2ab + 6b ab 3b

b a 9

a (b + c ) 2(b + c )
2

b c

3b
2

b a 9

3
2

a 9

(a 2)(b + c ) = a 2 , b ;
(b c )(b + c ) b c

3 2
= 10 .
5,6 5,7

 4
7
1. )
2. )

)
3.
4.

4
3

3a b

; )

a (a + b )
a

x 3y
a+b
1
=
.
; )
(x 3 y )(x + 3 y ) x + 3 y
a

3x 2 3x 2 3x
3x
ab + b 2 + a 2 ab a 2 + b 2
; )
= 2
;
=
b(a b )
b(a b )
x 2 1
x 1
2 y 2 2 y 2 + 16 y 16 y
=
.
y 8
y 8

2a 2 18 a 2 3a a 2 + 3a

18

(a 3) (a + 3)
(a 3)2 (a + 3)
4(x y ) + a (x y ) (a + 4 )(x y ) a + 4
, ;
=
=
(x + y )(x y ) x + y
x2 y2
2

2+4
6
=
= 12 .
7,3 7,8 0,5

1
8
1. ) ( 3) ( 7) 2 (3 5) = 2 7 3 + 21 62 + 10 = 52 + 21;
) 4 ( 2) ( 4) 2 = 42 8 2 + 8 16 = 32 16;
) 2 (m + 1) 2 4m = 2m2 + 4m + 2 4m = 2m2 + 2.
2. ) 3 9 = (2 9) = ( 3) ( + 3);
) 52 10ab 5b2 = 5 (a2 + 2ab + b2) = 5 (a + b) (a + b).
3. (2 2) 2 2 ( + 3) ( 3) + 2 (22 + 5) = 4 43 + 42 4 + 92 +
+ 43 + 10 = 132 + 10.
4. ) 164 81 = (42 9) (42 + 9) = (2 3) (2 + 3) (42 + 9);
) 2 2 = ( ) ( + ) ( + ) = ( + ) ( 1).
5. 2 4 + 9 = 2 4 + 4 + 5 = ( 2) 2 + 5 > 0.

2
8
1. ) 2 ( 3) 3 ( + 5) = 22 6 32 15 = 2 21;
) ( + 7) ( 1) + ( 3) 2 = 2 + 7 7 + 2 6 + 9 = 22 + 2;
) 3 ( + 5) 2 32 = 32 + 30 + 75 32 = 30 + 75.
2. ) 3 16 = (2 16) = ( 4) ( + 4);

167

) 3a2 6ab + 3b2 = 3 (a2 2ab + b2) = 3 (a b) (a b).

3. (3 2) 2 2 ( 2) ( + 2) + 2 (7 + 32) = 92 63 + 4 4 +
+ 42 + 14 + 63 = 132 + 14.
4. ) 814 1 = (92 1) (92 + 1) = (3 1) (3 + 1) (92 + 1);
) 2 2 6 9=2 (2 + 6 + 9)=2 ( + 3)2 = ( 3) ( + + 3).
5. 2 + 4 9 = (2 4 + 4 + 5) = ( ( 2) 2 + 5) < 0.

1
8
1. )

b(a b )(a + b )

a b
;
b

(a + b )b 2
2(x y )
2y
y2

=
;
y
(x y )(x + y ) x + y

8a 6
b3

8a 6
b3

) 33.

2.

a 2 a 2 + b 2 a b b 2 (a b )
b

=
=
.
(a b )a
2b
2ab(a b ) 2a

3.

(b 4)2 4(b + 3) = 2(b 4) ,

2(b + 3) (b 4 )(b + 4 )
b+4
b 3; 4; 4
2(2,4 4 )
3,2
=
= 0,5 .
2,4 + 4
6,4

x 2 + 2 xy + y 2 4 xy y 2 2 xy + x 2 + 4 xy (x y )2 (x + y )2

=
4.
=

(y x)
x+ y
yx
x+ y

= ( + ) ( ) = 2 2.

2
8
1. )

168

x+ y
x2
x

= ;
x a(x + y ) a

(a b )(a + b )
b

9 x10
y12

b
a b
3(a c )
1
3
=
; )

=
.
a (a + b )
a
a
(a c )(a + c ) a(a + c )

2.

x + y x + y (x y )(x + y ) 2 y 2

= 2 = .
(x y )(x + y )
y
y2
y

3.

(a 3)(a + 3) 4(a + 4 ) = 2(a 3) , 4; 2(1,8 3) = 2,4 = 0,5 .

2(a + 4 )
1,8 + 3
4,8
(a + 3)2 a + 3

1
y2
2 xy
1
2 x2 y2
x2
4. 2 + 2 + 2
=
+
+ 2
=
2
2
2
2
2
x

xy
y
x y
x y
x y2

x y
=

x 2 + 2 xy + y 2
x2 y2

( x + y )2 = x + y .
(x y )(x + y ) x y
3

8
1. ) 2 (1 + ) ( 2) ( + 4) = 2 + 22 2 4 + 2 + 8 = 2 + 8;
) ( + 2) 2 2 ( + 2) = 2 + 4 + 4 22 4 = 2 + 4;
) 30 + 3 ( 5) 2 = 30 + 32 30 + 75 = 32 + 75.
2. ) 4 3 = (4 2) = (2 ) (2 + );
) 2 + 2 + = (2 + 2 + 1) = ( + 1) 2 = ( + 1) ( + 1).
3. (b2 + 2b) 2 b2 (b 1) (b + 1) + 2b (3 2b2) = b4 + 4b3 + 4b2 b4 + b2 +
+ 6b 4b3 = 5b2 + 6b.
1 4
1
1

y = 4 y 2 4 + y 2 =
81
9
9

1
1
1 2

= 2 y 2 + y 4 + y .
3
3
9

4. ) 16

5. 2 2 + 12 = 2 2 + 1 + 11 = ( + 1) 2 + 11 > 0.

4
8
1. ) 5 (2 ) + 6 ( 7) = 10 52 + 62 42 = 2 32;
) (b 3) (b 4) (b + 4) 2 = b2 4b 3b + 12 b2 8b 16 =
= 15b 4;
) 20 + 5 ( 2) 2 = 20 + 52 20 + 20 = 52 + 20.
2. ) 25 3 = (25 2) = (5 ) (5 + );
) 42 + 8 42 = 4 (2 2 + 2) = 4 ( ) ( ).
3. (3 + 2) 2 2 ( 5) ( + 5) + 2 (8 32) = 92 + 63 + 4 4 +

169

+ 252 + 16 63 = 342 + 16.

16

4
4
2
2
4
b 4 = b 2 + b 2 = b + b + b 2 ;
81
9
9
3
3
9

) 2 2 + 4 4 = 2 (2 4 + 4) = 2 ( 2) 2 =
= ( + 2) ( + 2).

4. )

5. 2 + 2 5 = (2 2 + 5) = (2 2 + 1 + 4) = ( ( 1) 2 + 4) < 0.

3
8
1. )

x y
x2
x

=
;
(x y )(x + y ) x x + y

(a b )(a + b ) = a b ;
a

3(a + b )
3a
a2

32c15
y5

4x 2 y
y

;
4x 2
.
y

y 2 y 2 + 2 xy x 2 y x x(2 y x )( y x ) 2 y x
.
=

=
y( y x )
x
yx( y x )
y

2.

2(x 2 )

(x 6 )(x + 6 ) = x 6 , 2; 6;
8(x 2 )
4(x + 6 )
(x + 6 )

3.

1,5 6
4,5
=
= 0,15 .
4(1,5 + 6 )
30
a 2 16a + 64 + 32a a 2 + 16a + 64 32a
=

a 8
a +8

4.

a 2 + 16a + 64 a 2 16a + 64 (a + 8)2 (a 8)2

=
=

a 8
a +8
(a 8)(a + 8)
= ( + 8) ( 8) = 2 64.
=

4
8
1. )

5(a + b )
2

b
5
= ;
a+b b

b
( y 1)( y + 1) = y + 1 ;
y
)

x( y 1)
y
x

170

)
)
2.

3.

4a 6
b8

(a x )(a + x )
1

a
a 2 ax
=
.
2(a + x )
2

3y + 6 3y + 6 y 2 4
12 y 2 4
12
=
= 2

.
( y 2)( y + 2 ) y + 4 y 4 ( y + 4 ) y + 4

(c 5)2 4(c + 2 ) = 2(c 5) , 2; 5;

2(c + 2 ) (c 5)(c + 5)
c+5
2(7,5 5 )
5
=
= 0,4 .
7,5 + 5

12,5

1
1
2(a b ) a 2 b 2
1
1
2 a 2b 2

4. 2 + 2
=
+

=
ab(a b ) a 2 b 2 a 2 b 2 ab a 2 b 2
b
a
=
=

b2
a2 b2

a2
a2 b2

2ab
a2 b2

(a b )2 = a b .
(a b )(a + b ) a + b

a 2 2ab + b 2
a2 b2

1
9

4 x + y = 3
;
1.
6 x 2 y = 1

y = 3 4x
;

6 x 6 + 8 x = 1

y = 1
.

x = 0,5

2. , ;
x = 8 y
x + y = 8
;
;

2000 x + 3000 y = 19000 16000 2000 y + 3000 y = 19000

x = 5
x = 8 y
().
;

1000 y = 3000 y = 3
6 x + 4 y + 9 = 4 x + 21 2 x + 4 y = 12
;
;
3.
2 x + 10 = 3 6 x 5 y 8 x + 5 y = 7
2 x + 4 y = 12 x = 6 2 y x = 4
;
;
.

y = 5
11 y = 55 y = 5

8 x 16 y = 48
;

8 x + 5 y = 7

171

8 = k 3 + b
;
4. :
1 = 4k + b

b 4k = 1
;

7 = 7 k

b = 5
; = + 5.

k = 1

3
7

y= x
3 x 2 y = 7
2
2.
;
5.
6 x 4 y = 1 y = 3 x 1

2
4
(k1 = k2) , b1 b2,
, , .

2
9

3 x y = 7
;
1.
2 x + 3 y = 1

y = 3x 7
;

2 x + 9 x 21 = 1

y = 1
.

x = 2

2. / , /
.
:
2 y + x = 40 2 y + y + 4 = 40 y = 12
(/).
;
;

x = 16
x = y + 4
x y = 4

6 x 2 y 5 = 2 x 3 y
;
3.
5 x + 2 y = 4 y + 16

4 x + y = 5
;

x 2 y = 11

y = 7
8 y 44 + y = 5 7 y = 49
.
;
;

x = 2 y 11 x = 3
x = 2 y 11
b = 5k
0 = 5k + b
;
;
4. :
21 = 2k + b 7 k = 21
b = 15
; = 3 + 15.

k = 3
y = 5 x 11
5 x y = 11
;
;
5.

10
x
+
2
y
=

22
y = 5 x 11

, , , .

172

 1
9
1. ) = 10,5;
) 3 = 3 + 18;
= 7;
) 3 = 3 (5) + 18;
3 = 3 , , (5; 3);
2. )

3.

6
= -0,5
=3
3
1

1
-3

-2

= 2 + 6

) = 9.
4. y = 5x + b;
52 = 50 + b;
b = 2.
5. y = kx + b;
k = 7;
y = 7x + b;
.. = 7x + b (0; 0) , :
0 = 7 0 + b; b = 0; = 7.

2
9
1. ) = 5 2,5 + 10 = 2,5;
) 5 = 5 + 10;
5 = 15;
= 3;
) 5 = 5 3 + 10;

173

5 = 5 , , (3; 5);
2. )

3.

= 0,5
1
0

1
1

= -2 + 6

= -4
-4

) = 4.
4. y = kx 12; 7 = 15k 12; k =

1
.
3

5. y = kx + b; k = 8; y = 8x + b;
.. = 8x + b (0; 0) , :
0 = 8 0 + b; b = 0;
= 8.

3
9

4 x + 3 y = 2 16 y 36 + 3 y = 2 y = 2
.
;
;
1.
x = 1
x = 4 y 9 x = 4 y 9
2. .
2 x + 4 y = 70 50 2 y + 4 y = 70
;
;
:
x = 25 y
x + y = 25

6 x + 3 y 26 = 3x 2 y 3 x + 5 y = 26
.
;
3.
3 x + 3 y = 10
15 x + 3 y = 3x + 5

174

y = 10
.

x = 15

3 x + 5 y = 26 3 x = 26 5 y x = 5
;
;
3.

y = 2
8 y = 16
y = 2

9 = 10k + b 9 = 10k + b b = 1
; = + 1.
;
;
4. :
k = 1
16 = 16k
7 = 6k + b
5
8

y = 3 x 3
.

Y = 5 x 8

3
9
(k1 = k2) , b1 b2, , ,
.
5 x 3 y = 8
;
5.
15 x 9 y = 8

4
9

3 x 2 y = 16 12 y 12 2 y = 16 y = 2
.
;
;
1.
x 4
x + 4 y = 4 x = 4 y 4
2. . , .
:
15 x + 10 y = 35000 15 x + 10 x + 10000 = 35000
;
;

y = x + 1000
y x = 1000

25 x = 25000 x = 1000
().
;

y = x + 1000 y = 2000
4 x y 24 = 10 x 4 y 6 x + 3 y = 24
;
;
3.
x + 2 y = 6
3 y 2 = 4 x + y
3 y 6 x = 24
;

x = 6 2 y

3 y 36 + 12 y = 24
;

x = 6 2 y

11 = 2k + b
;
4. :
4 = 12k + b

y = 4
.

x = 2

11 = 2k + b
;

7 = 14k

b = 10
; = 0,5 + 10.

k = 0,5
175

 y = 4x 7
4 x y = 7
;
;
5.
2
y
+
14
=
8
x
y = 4x 7

, , , .

3
9
1. ) = 4 (2,5) 20 = 30;
) 4 = 4 20;
4 = 24;
= 6;
) 28 = 4 (2) 20;
28 = 29 , , (2; 28);
2. )

3.

= -2

= 2 - 6

2
1

1
0

-1 0

= -5

-6

) = 3.
4. y = 3x + b (15; 40) ,
: 40 = 3 15 + b; b = 5.
5. y = kx + b;
k = 6;
y = 6x + b;
.. y = 6x + b (0; 0) ,
0 = 6 0 + b; b = 0; y = 6x.

176

-5

 4
9
1. ) = 4 2,5 18 = 28;
) 2 = 4 18;
4 = 20; = 5;
) 20 = 4 2 18;
20 = 26 , , K (2; 20);
2. )

3.

= 2

=4

= -2 6
2
1

1
-3

-6

) = 2.
4. y = kx + 15 (8; 11) , ,
11 = 8k + 15; 8k = 4;
k = 0,5;
5. y = kx + b = 12 3, , k = 12;
y = 12x + b;
.. = 12 + b (0; 0) , :
0 = 12 0 + b; b = 0; = 12.

1
10

5 x + 3 y = 4 5 x + 6 x + 15 = 4 11x = 11 x = 1
.
;
;
;
1.
y = 2x + 5 y = 3
2 x y = 5 y = 2 x + 5
177

2. 1000 . 1500 .
:
x + y = 30
x = 30 y
;
;

1000 x + 1500 y = 39000 30000 1000 y + 1500 y = 39000

x = 12
x = 30 y
.
;

500
y
=
9000
y = 18

8 x 6 y = 3
;

20 x 14 y = 65
1

40 x 30 y = 15 8 x + 6 y = 3
x = (3 6 y ) x = 1,5
;
;
.
;
8

40 x 28 y = 130 58 y = 145 y = 2,5

y = 2,5

10 8 x 20 = 6 y 13
;
3.
4 y 63 = 20 x 10 y + 2

26 = 3k + b
;
4. :
22 = 5k + b

26 = 3k + b
;

48 = 8k

b = 8
; = 6 + 8.

k = 6

y =

y =

2
1
x
7
7
.
2
5
x
7
14
(k1 = k2) , (b1 b2)
, , , .

2 x 7 y = 1
;
5.
4 x 14 y = 5

2
10

x = 6 y + 20
;
1.
24 y + 80 + 2 y = 2

x = 6 y + 20 x = 2
.
;

26 y = 78 y = 3

2. / , / .
x = 5
x + 2 y = 19 x + 2 x + 4 = 19 3 x = 15
(/).
;
;
;
:
y
=
x
+
2
y
=
x
+
2
y

x
=
2
y = 7

15 x + 9 y 6 = 2 x + 11 13 x + 9 y = 17
;
;
3.
4 x 15 = 11 8 x + 2 y 12 x 2 y = 26
12 x 2 y = 26 6 x y = 13
;
;

x + 11 y = 9 x = 9 11 y
178

 54 66 y y = 13
;

x = 9 11 y

67 y = 67
;

x = 9 11 y

6 = 4k + b
;
4. :
12 = 8k + b

y = 1
.

x = 2

4k + b = 6
;

8k b = 12

4k + b = 6 b = 8
; = 0,5 8.
;

k = 0,5
12k = 6
3
2

y = 5 x + 5
;

y = 3 x + 2

5
5
, , ,
.

3 x + 5 y = 2
;
5.
6 x + 10 y = 4

3
10

2 x 3 y = 12 2 4 y 3 y = 12
;
;
1.
x = 1 2 y
x + 2 y = 1
y = 2
7 y = 14
.
;

x = 1 2 y x = 3
2. , .
:
3 x + 5 y = 290 930 21 y + 5 y = 290
;

x + 7 y = 310 x = 310 7 y

16 y = 640
;

x = 310 7 y

y = 40

x = 30

6 x + 3 = 8 x 6 y + 12
;
3.
4 x 6 y 4 x = 2 y 8

() 1-.
() 2-.
2 x + 6 y = 9 2 x = 6 y 9 x = 1,5
.
;
;

y =1
y = 1
8 y = 8

2 k + b = 1
1 2k + b
;
;
4. :
10 = 4k + b 4k b = 10

179

2 k + b = 1
;

6k = 9

b = 1 2k
;

k = 1,5

b = 4
;

k = 1,5

= 1,5 + 4.
3

y = 2 x + 2
;

y = 3 x + 5

2
3
(k1 = k2) ,
(b1 b2) , , , .
3 x + 2 y = 4
;
5.
9 x + 6 y = 10

4
10

5 x + y = 14
;
1.
2 x 3 y = 9

y = 14 5 x
;

2 x 42 + 15 x = 9

y = 14 5 x
;

17 x = 51

y = 1
.

x = 3

2. , .
x = 9 y
x + y = 9
;
;
:
2 x + 3 y = 23 18 2 y + 3 y = 23
x = 4

y =5

4 y + 20 = 6 x 8 y 4
;
3.
16 5 x 2 y = 3 x 2 y
7 = 6k + b
;
4. :
11 = 2k + b

12 y 6 x = 24
;

8 x = 16

12 y = 6 x 24
;

x = 2

y = 1
.

x = 2

6k + b = 7
;

2k b = 11

b = 7 6k b = 10
; = 0,5 + 10.
;

k = 0,5
8k = 4
1
2

y = 7 x 7
;

y = 1 x 2

7
7
, , ,
.

x 7 y = 2
;
5.
3 x 21 y = 6

180


1
1
1. ( + 6) 2 2 (3 2) = 2 + 12 + 36 6 + 42 = 52 + 6 + 36.

x = 1
5 x 2 y = 11 5 x 8 x + 8 = 11 3 x = 3
.
;
;
;
2.
y
=
4
x

4
y
=
4
x

4
4
x

y
=
4
y = 8

3. )

= 2 2

1
0

-2

) 20 = 2 (10) 2;
20 = 22 , , ;
(10; 20).
4. ) 2a4b3 2a3b4 + 6a2b2 = 2a2b2 (a2b ab2 + 3);
) 2 3 3 2 = ( ) ( + ) 3 ( + ) = ( + ) ( 3).
5. / .
. .. , (
2) /. 3 , 2 . :
2 3 + ( 2) 2 = 30;
2 4 + 6 = 30;
2 = 28;
= 14 (/) .

2
1
1. ( 2) 2 ( 1) ( + 2) = 2 4 + 4 2 2 + + 2 = 5 + 6;

181

3 x + 5 y = 12 6 y 21 + 5 y = 12
;
;
2.
x 2 y = 7 x = 2 y 7
3. )
= 2 + 2

11 y = 33
;

x = 2 y 7

y = 3
.

x = 1

2
1
0

) 18 = 2 10 + 2;
18 = 18 , , (10; 18).
4. ) 333 + 324 62 = 32 (2 + 2 2);
) 2 + 2 b2 2b = 2 (a b) + (a b) (a + b) = (a b) (2 + a + b).
5. / . ( + 28) / . 0,5 , 0,5 + 0,5 = 1 . :
1 + 0,5 ( + 28) = 32;
+ 0,5 + 14 = 32;
1,5 = 18;
= 12 (/) ;
12 + 28 = 40 (/) .

1
3
1.

8a(a + 4 )
a 2 4a a 2 4a a + 4
8
8
=
=
=

.
(a + 4 )(a 4)
a
a (a + 4 )(a 4 )
a4 4a

2. )

182


= -0,5

1
-2

) = 2.
3.

4
x 9
x=
; 8 = 5 ( 9); 8 = 5 45; 3 = 45; = 15.
5
2

3 x 2 y = 5 2 x 2 y
;
4.
4 x 4 y = 2
a
2ac
5.
+
a c (a c )2

x = 1
5 x = 5
.
;

4
y
=
4
x
+
2
y = 1,5

4ac a 2 2ac c 2

a+c

a 2 ac + 2ac (a c )2 a (a + c )(a c )2

=
( 1) = a .
a+c
(a c )2
(a + c )(a c )2

2
3
1.

a
a2 b2 + b2
a
b(a b ) b
:
=

= .
ab
b(a b )
ab
a
a2

2. )

= 2

2
1
0

) = 2,5
183

2x
x 5
4 =
4 5 12 = 2; = 17.
3
3

3.

3 x + 3 y = 6
x + y = 2
4.
;
;
6 5 x + 5 y = 8 x = 2 y 13 x 7 y = 6
x = 2 y x = 1
x = 2 y
.
;
;

26

13
y

7
y
=
6
20 y = 20 y = 1

a 2 2ab + b 2 + 4ab 4a 2 2a 2 2ab (a + b )2 2a (a b )

=
= 2a .
a b
(a + b )2
(a b )(a + b )2

5.

3
1
1. 2 (2 + 3) ( + ) 2 = 42 + 6 2 2 2 = 32 + 4 2.

x = 8 y 6 x = 2
x + 8 y = 6 x = 8 y 6
.
;
;
;
2.
5 x 2 y = 12 40 y 30 2 y = 12 42 y = 42 y = 1
3. )

= 2 + 2

2
-1 0

) 20 = 2 10 2;
20 = 22 , , ; (10; 20)
4. ) 332 342 + 92 = 32 (2 2 + 3);
) 2 2 + 2 + 2 = 2 ( + ) + ( ) ( + ) = ( + ) (2 + ).
5. / . ..
, ( 2) /. . 1,5 , 1,5 + 0,5 = 2 .
: 1,5 2 + 2 ( 2) = 35; 3 + 2 4 = 35; 2 = 36; = 18 (/).

184

 3
3
a +ba +b a b
2b(a b )
2

=
=
.
1.
(a b )(a + b ) b b(a b )(a + b ) a + b
2. )

y = 0,5

1
0

) = 2.
3.

x 3 2x
=
; 7 ( 3) = 4; 7 21 = 4; 3 = 21; = 7.
2
7

14 3 x + 3 y = 5 y x 2 y + 2 x = 14 x + y = 7
;
;
;
4.
x + y = 4
x + y = 4
x + y = 4
, .
5.

10 x + x 2 5 x 20 x x 2 10 x 25 x(x + 5 ) (x 5)2

=
=
x+5
( x 5 )2
(x + 5)(x 5)2
=

x(x + 5)(x 5)2

(x + 5)(x 5)2

= x .

4
3
1.

y
x2 x2 + y2
y
x (x + y ) x
:
=

= .
x+ y
x (x + y )
x+ y
y
y2

2. )

185


= -2
4

1
-2

) = 2,5.
3x
x+3
1 =
; 3 2 = + 3; 2 = 5; = 2,5.
2
2

3.

x y = 2
;
4.
3 x 7 y = 20 2 x 2 y

x y = 2
;

5 x 5 y = 20

x y = 2
; , .

x y = 4
a 2 + 6a + 9 12a 2a 2 6a 4a 2 (a 3)2 2a (a + 3)

=
a+3
a+3
(a 3)2
(a 3)2

5.

2a (a + 3)(a 3)2

(a + 3)(a 3)2

= 2a .

1
2
1. ) 3a2b (5a3b) = 15a5b2; ) (22) 3 = 863.
2. 3 5 (2 + 1) = 3 (3 2); 3 10 5 = 9 6; = 14.
3. ) 2 62 = 2 ( 3); ) 3 4 = (2 4) = ( 2) ( + 2).
4. , ( + 2) , 2 . :
+ + 2 + 2 = 50; 4 = 48;
= 12 () ; 12 + 2 = 14 () ; 2 12 = 24 () .
5. ( + ) ( ) b (2a b) (a b + c) (a b c) = 0;
a2 c2 2ab+ b2 (a b) 2 + c2=a2 c2 2ab + b2 a2 + 2ab b2 + c2=0.
6. (; ) , = 5 8, ..

186

 = 5 8; 6 = 8; a =

4 4 4
; ; .
3 3 3

4 4
: ; .
3 3

2
2
1. ) 22 335 = 647; ) (4ab3) 2 = 16a2b6.
2. 4 (1 5) = 9 3 (6 5); 4 20 = 9 18 + 15;
2 = 20; = 10.
3. ) a2b ab2 = ab (a b); ) 9 3 = (9 2) = (3 ) (3 + ).
4. , ( 10)
, ( 10 5) = ( 15) .
: + 10 + 15 = 50; 3 = 75;
= 25 () ; 25 10 = 15 () ;
25 15 = 10 () .
5. ( ) ( + ) ( + ) (2 ) = 0;
2 2 ( ) 2 + 2 2 + 2 = 2 2 2 + 2 2 + 2 2 +
+ 2 = 0.
6. (; ) , = 3 + 8,
.. = 3 + 8; 2 = 8; = 4; (4; 4);
: (4; 4).

3
2
1. ) 75 (242) = 1457; ) (5a3b) 2 = 25a6b2.
2. 3 4 (1 6) = 2 (3 + 4); 3 4 + 24 = 6 + 8; 18 = 9; = 0,5.
3. ) 3a2 9ab = 3a (a 3b); ) 3 25 = (2 25) = ( 5) ( + 5).
4. ., ( + 300) . , 3 .
. :; + + 300 + 3 = 600;
5 = 300; = 60 (.) ;
60 + 300 = 360 (.) ;
3 60 = 180 (.) .
5. (a x) (a + x) b (b + 2x) (a b x) (a + b + x) = 0;
a2 x2 b2 2bx (a (b + x)) (a + (b + x)) = a2 x2 b2 2bx a2 +
+ (b + x) 2 = a2 x2 b2 2bx a2 + b2 + 2bx + x2 = 0.

187

6. (; ) , = 3 + 10, ..
= 3 + 10; 2 = 10; = 5 (5; 5);
: (5; 5) .

4
2
1. ) 747 (32) = 2159; ) (2a5b) 3 = 8 a15 b3 = 8a15b3.
2. 2 (3 2) = 3 4 (1 + 3); 6 4 = 3 4 12; 5 = 10; = 2.
3. ) 22 + 42 = 2 ( + 2);
) 100 3 = (100 2) = (10 ) (10 + ).
4. , ( + 5) ( + 5 15) . :
+ + 5 + 10 = 100;
3 = 105; = 35 () ;
35 + 5 = 40 () ;
35 10 = 25 () .
5. ( + ) ( ) ( + ) ( + ) ( 2) = 0;
2 2 ( ( )) ( + ( )) 2 + 2 = 2 2 2 + ( ) 2
2 + 2 = 2 2 2 + 2 2 + 2 2 + 2 = 0.
6. (; ) , = 2 + 15,
.. = 2 + 15; 3 = 15; = 5; (5; 5). : (5; 5).

1
1. , ,
.
) 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92;
1 + 4 + 9 + 6 + 5 + 6 + 9 + 4 + 1 = 2 (1 + 4 + 9 + 6) + 5 = 45
. : 5.
) 942 + + 1942 = 942 + (1902 + 1912 + + 1942 + 952 + + 992) +
+ (1002 + 1012 + + 1092) + + (1802 + + 1892).
,
) , .. . 10 , ..
(10 5 = 50) . 942 6.
, 6 + 0 = 6 . : 6.
2. 2 $ 89 289 . 289 = 17 17 = 289 1
n k .
.. n k = 289 = 17 17 = 289 1;

188

 n k n, k > 1. , n = k = 17;
: 17 .
3. V1 = 600 : 6 = 100 (/) ;
V2 = 600 : 3 = 200 (/) ;
) t .
: 100t + 200t = 600; 300t = 600; t = 2 ();
: 2 .
) 6 ., 6 . 2 , , . : 6 .
4. ) . .

) .
, 101;
1 100;
2 99;
3 98;
.
50 51;
51 50;
.
99 2;
100 1.
5. : .

2
1. ) 12 + + 92 5.
1 .
) 972 + + 1972 = 972 + (1902 + + 1972 + 982 + 992) + (1002 +
+ 1092) + (1102 + + 1192) + + (1802 + + 1892)

189

 ,
) , .. . 10 ,
, (5 10 = 50) . 972 9.
.. 9 + 0 = 9 .
: 9.
2. n k .
.. n k = 361 = 19 19 = 361 1
n k , n < 200, n > 1
, n = k = 19
: 19 .
3. ) , 2 , . , . 2 . ,
. , 4 .
, 1 , 2 .
: 4 .
) ) , 4
. ,
, 10-
36 .:
2 4 ;
3 8 ;

10 36 .
: 36 .
4. ) . , 28 28 28
, 8 ,
, ,
28 .
) . , 65
1 64;
2 63;
3 62;

30 35;

32 33;
33 32;

64 1.
5. :
.

190

 .
1
1. 8
(2, 3, 4, 5, 6, 7, 8, 9) , . 9 , , . , () : 8 9 = 72;
: 72 .
2. ) ( + 1) (14 13 + 12 11 + + 2 + 1) = 15 + 14 14 13 +
+ 13 + 12 + + + 1 = 15 + 1;
) = 1989, ) :
198915 + 1 = (1989 + 1) (198914 + 1) = 1990 (198914 + 1) ,
1990 995, 1990 = 2 995.
3. 8 7 , ;
5 8 , ,
3 .
, 8 + 7 3 = 29 .
4. ) 16 , ,
4 16 4 = 64;
) : (1; 1) (1; 15).
(2; 1) (2; 15);

(15; 1) (15; 15);

.. 15 15 = 225 .
) 4 : (0; 0) ; (1; 2) ; (2; 4) ; (3; 6);
, , ..
(0; 0) (3; 6);

0 = k 0 + b ;
6 = 3k + b

b = 0 ; = 2;
k = 2

, , = 2
, .. 0 3.
) ,

0 = k 0 + b ;
61 = 31k + b

b = 0
61
x;
k = 61 , .. y =
31

31

(a; b) , b ,
, 0 31 ( ) .
: b =

61
a;
31

D (61, 31) = 1, 31 . , b
= 0, b = 0, = 31, b = 61;
: 2 , (0; 0) , (31; 61) .
191

 2
1. 5
(1, 3, 5, 7, 9) , 5 ( ) . ,
5 5 = 25.
: 25.
2. ) ( 1) (14 + 13 + + 1) = 15 14 + 14 13 + + 1=15 1
) ) = 1989, :
198915 1 = (1989 1) (198914 + + 1) = 1988 (198914 + + 1) ,
1988 994, .. 1988 = 2 994.
3. 7 3 , ;
5 4 , ;
2 , 1 , .. 2
, .. , , 50% .
4. ) = 100
(2; 100) , .
= 100 (HOD (2;
100) = 2) , > 5.
.. ,
.
: .
) , = 5, (; 5) .
,
5 . , 2.
, (2; 10) .
: .
) (; ) , .
= 2, = 2, 4, 6, 8, 10, 12, 14;
= 4, = 2, 6, 10, 14;
= 6, = 2, 4, 8, 10, 14;
= 8, = 2, 6, 10, 14;
= 10, = 2, 4, 6, 8, 12, 14;
= 12, = 2, 10, 14;
= 14, = 2, 4, 6, 8, 10, 12;
: 35.
: 35.
) , , (0; 0)
(31; 61) . , HOD 2, (0; 0) (31; 61) . .
: .

192