Laws of Exponents
1. 32 ⋅ 35 = 37
2. 7 3 ⋅ 7 4 = 7 7
3. 114 ⋅117 = 1111
4. 2 5 ⋅ 2 3 ⋅ 2 4 = 212
5. 4 6 ⋅ 4 3 ⋅ 4 5 = 414 = (2 2 )14 = 2 28
6. 7 ⋅ 7 3 ⋅ 7 5 = 7 9
7. 32 ⋅ 2 3 = 32 ⋅ 2 3
8. 5 3 ⋅ 2 4 ⋅ 5 4 ⋅ 2 2 = 5 7 ⋅ 2 6
9. 4 ⋅ 35 ⋅ 4 3 ⋅ 3 = 4 4 ⋅ 36 = (2 2 )4 ⋅ 36 = 2 8 ⋅ 36
10. x 3 ⋅ x 2 = x 5
11. x 6 ⋅ x 4 = x10
12. x 3 ⋅ x 4 ⋅ x 2 = x 9
13. x 3 ⋅ y 2 = x 3 ⋅ y 2
14. x 4 y 5 ⋅ x 2 y 3 = x 6 y 8
15. x 4 y 2 z 3 ⋅ x 2 y 3 z 5 = x 6 y 5 z 8
16. 3x ⋅ 5x = 3 ⋅ 5 ⋅ x 2
17. 4x 2 ⋅ 5x 3 = 2 2 ⋅ 5 ⋅ x 5
18. 3x 2 y 3 ⋅ 5x 4 y 2 = 3 ⋅ 5 ⋅ x 6 y 5
19. 32 x ⋅ 3x = 33 x 2
20. 4 3 x 2 ⋅ 4 2 x 3 = 4 5 x 5 = (2 2 )5 x 5 = 210 x 5
21. 5 3 x 4 y 2 ⋅ 5 2 x 2 y 6 = 5 5 x 6 y 8 22. 32 x ⋅ 4x = 32 ⋅ 2 2 ⋅ x 2
10
23. 4 2 x 3 ⋅ 5x 2 = (2 2 )2 ⋅ 5 ⋅ x 5 = 2 4 ⋅ 5 ⋅ x 5
( )
= 56
27. 7 3
( )
= x6
30. x 5
3
25. ( 3) = 33
26. 52
28. ( x ) = x 4
29. x 3
( )
32. x 2 y
2
4
( )
3
31. xy = x 3 y 3
(
)
(
)
34. x 5 y 3 z 4
37. 7x 3 y 2
4
4
4
= x 6 ⋅ x 4 = x10
( ) ( )
42. x 3 ⋅ x 5
(
3
= x 6 ⋅ x15 = x 21
) ⋅ ( x yz )
2
2
4
= x15
(
)
3
36. 5x 2
3
( )
= x12 y 3 z 6
)
3
(
= 7 6 x15 y12
( ) ( )
3
4
= x12 y 6
= 53 x 6
) ( )
2
)
4
= 516 x 24 z 4
= x12 ⋅ x 8 = x 20
43. x 2 y 3 ⋅ xy 2
= x 8 y 6 z 4 ⋅ x 8 y 4 z 4 = x16 y10 z 8
3
3
39. 54 x 6 y 3 z
41. x 4 ⋅ x 2
(
= 712
( )
)
38. 7 2 x 5 y 4
3
4
33. x 4 y 2
= 7 4 x12 y 8
(
( )
= x8 y 4
35. x 4 yz 2
( ) ()
44. x 4 y 3 z 2
(
4
= x 20 y12 z16
40. x 2 ⋅ x
2
2
10
24. 32 x 5 y 2 ⋅ 4 2 x 5 y 3 = 32 (2 2 )2 x y 5 = 32 ⋅ 2 4 ⋅ x y 5
3
= x 4 y 6 ⋅ x 3 y 6 = x 7 y12
( ) ( 4x )
45. 2x
3
2
2
= 23 x 3 ⋅ 42 x 4 = 23 x 3 ⋅ (22 )2 x 4 = 23 x 3 ⋅ 24 x 4 = 27 x 7
(
) (
)
(
) (
) (
(
) ⋅ ( 4x y ) = (3x y ) ⋅ ( 2 x y )
3
46. 3x 2 y ⋅ 32 xy 3
2
2
= 33 x 6 y 3 ⋅ 34 x 2 y 6 = 37 x 8 y 9
) (
3
)
2
3
47. 5xy 3 ⋅ 4x 2 y = 5xy 3 ⋅ 22 x 2 y = 52 x 2 y 6 ⋅ 26 x 6 y 3 == 26 ⋅ 52 x 8 y 9
48. 3x 3 y 2
4
2
2
4
3
(
) (
(
) ⋅ (6xy ) = ( 2
3
49. 52 x 3 y 4 z ⋅ 5x 4 y 2 z 2
50. 12x 3 y 2
4
)
4
3
4
2
2
2
4
2
= 34 x12 y 8 ⋅ 24 x 4 y 8 == 24 ⋅ 34 x16 y16
= 56 x 9 y12 z 3 ⋅ 54 x16 y 8 z 8 = 510 x 25 y 20 z11
2
⋅ 3x 3 y 2
) ⋅ ( 2 ⋅ 3xy )
4
3
= 28 ⋅ 34 x12 y 8 ⋅ 23 ⋅ 33 x 3 y 3 = 211 ⋅ 37 x15 y11
(10x y z ) ⋅ ( 20xy z ) = ( 2 ⋅ 5x y z ) ⋅ ( 2
51.
4
3
2 2
4 2
4
4
2 2
3
2
⋅ 5xy 4 z 2
)
4
= 23 ⋅ 53 x12 y 6 z 6 ⋅ 28 ⋅ 54 x 4 y16 z 8 =
211 ⋅ 57 x16 y 22 z14
(
52. 15x 3 y 2
) (75x y ) = (3⋅ 5x y ) (3⋅ 5 x y )
3
4
2
(
58. −7x 4 y −5
(
57. 5x −2 y 3
)
3
= −7 3 x12 y −15 = −
(
)
(
) (x y )
62. x 2 y −3
5
2
4
= 33 ⋅ 53 x 9 y 6 ⋅ 34 ⋅ 58 x 8 y 4 = 37 ⋅ 511 ⋅ x17 y10
y2
55. x y =
x
−1
y
x3
60. 5x −3 y 4
3
2
1
54. (xy) =
xy
1
53. 3 =
3
−1
56. x −3 y =
3
2
−2
= 5−2 x 6 y −8 =
−1
2
3
7 3 x12
y15
x6
52 y 8
= x 4 y −6 ⋅ x −3 y 6 = x
)
2
−1
= 52 x −4 y 6 =
52 y 6
x4
(
)
(
)
59. 11x −2 y −3
61. 3−3 x −2 y 4
2
−3
2
= 112 x −4 y −6 =
112
x4 y6
= 39 x 6 y −12 =
39 x 6
y12
(3
−2
63.
x −4 y 2 z 3
) (9x y z )
2
2
3
3 −2
3−4 x −2 y13 ⋅ 36 = 32 x −2 y13 =
= 3−4 x −8 y 4 z 6 ⋅ 93 x 6 y 9 z −6 = 3−4 x −8 y 4 z 6 ⋅ (32 )3 x 6 y 9 z −6 =
32 y13
x2
(27x −4 )2 (81−2 x)3 (9x −2 )−2 = (33 x −4 )2 ([34 ]−2 x)3 (32 x −2 )−2 = (33 x −4 )2 (3−8 x)3 (32 x −2 )−2 =
64.
36 x −8 ⋅ 3−24 x 3 ⋅ 3−4 x 4 = 3−22 x −1 =
(
65. 125−2
1
22
3 x
) (625 ) (3125) = ([5 ] ) ([5 ] ) (5 ) = (5 ) (5 ) (5 )
3
3
4
−2
66.
x2
1
= 2
4
x
x
69.
27x 6
33 x 6
3x 2
=
=
144x 4 24 ⋅ 32 x 4
24
3 −2
67.
3
4 3
4
5 −2
−6
3
x4
1
= 6
10
x
x
68.
70.
12x 3 y 5 z 2
22 ⋅ 3x 3 y 5 z 2
x
71.
= 2
= 2 2
2 7 4
2 7 4
60x y z
2 ⋅ 3⋅ 5x y z
5y z
12
4
5 −2
= 5−18 ⋅ 548 ⋅ 5−10 = 520
3x 4
3x 4
x2
=
=
2
6x 2 2 ⋅ 3x 2
x4 y7
= xy 5
3 2
x y
x −4
1
1
72. 2 = 2 4 = 6
x
x x
x
73.
x7
= x 7 x 3 = x10
−3
x
74.
x 2 y −5 x 2 x 4 y 8
=
= x6 y3
−4 −8
5
x y
y
75.
45x −3 y 6 32 ⋅ 5x 2 y 6 y 3 5y 9
=
= 2
81x −2 y −3
34 x 3
3 x
76.
75x 4 y −5 z 2
3⋅ 52 x 4 x 3 y 7 z 2 3⋅ 5x 7 y 2
=
=
245x −3 y −7 z 9
5 ⋅ 72 y5 z 9
72 z 7
(x y )
77.
( x y)
2
3
2
3
4
(5x )
78.
3
x4 y6
y3
= 12 3 = 8
x y
x
15x 4
(16x y ) = ( 2 x y )
79.
4
2
5
4
4
2
24 ⋅ 3x 9 y 7
48x 9 y 7
(81x y ) = (3 x y )
80.
5
3
144x 4 y
5
2
4
5
3
2
=
220 x 20 y10 216 x11 y 3
=
3
24 ⋅ 3x 9 y 7
38 x10 y 6
36 x 6 y 5
=
=
24 ⋅ 32 x 4 y 24 ⋅ 32 x 4 y
24
2
=
52 x 6
5x 2
=
3
3⋅ 5 ⋅ x 4
81.
625x 6 y 7
54 x 6 y 7
=
(35x y ) (5 ⋅ 7x y )
5
4
5
4
(12x y ) = ( 2 ⋅ 3x y )
82.
( 20xy ) ( 2 ⋅ 5xy )
4
2
3
2
4
2
4
3
2
54 x 6 y 7
y2
=
55 ⋅ 75 x 20 y 5 5 ⋅ 75 x14
=
2
2
4
24 ⋅ 32 x 8 y 6
32 x 4
= 8 4 4 8 = 4 4 2
2 ⋅5 x y
2 ⋅5 y
(80x y z ) = ( 2 ⋅ 5x y z )
83.
(32x y z ) ( 2 x y z )
6
3
3
3 2
4
2
5 4
6
5
3
3
3 2
2
5 4
(120x z ) = ( 2 ⋅ 3⋅ 5x z )
84.
(90x z ) ( 2 ⋅ 3 ⋅ 5x z )
4
4 2
2 3
4
4 2
3
2
3
2 3
=
212 ⋅ 34 ⋅ 54 x16 z 8 29 ⋅ 5x10
=
23 ⋅ 36 ⋅ 53 x 6 z 9
32 z
(6x y ) (9x y ) = ( 2 ⋅ 3x y ) (3 x y )
(108x y )
(2 ⋅ 3 x y)
2
85.
3
212 ⋅ 53 x18 y 9 z 6 22 ⋅ 53 x12
=
210 x 6 y10 z 8
yz 2
=
3
3
2
4
2
4
2
3
2
2
3
3
4
2
4
2
23 ⋅ 33 x 6 y 3 ⋅ 34 x 6 y 8 23 ⋅ 37 x12 y11 x 4 y 7
= 8 12 8 4 = 5 5
28 ⋅ 312 x 8 y 4
2 ⋅3 x y
2 ⋅3
=
(10x y ) ( 25x y ) = ( 2 ⋅ 5x y ) (5 x y )
86.
( 45x y )
(3 ⋅ 5x y )
4
4
2
3
3
3
2
5
4
3
4
2
3
2
3
3
3
2
2
−4
−5
−4
3
−3
2
2
2
−4
2
2
−4
−5
2
=
3
( 2x y ) (8x y ) = ( 2x y ) ( 2 x y )
( 20x y )
( 2 ⋅ 5x y )
87.
−3
5
3
2
24 ⋅ 54 x16 y 8 ⋅ 54 x 6 y10 24 ⋅ 58 x 22 y18 24 ⋅ 55 x13 y 9
= 9 3 9 9 =
39 ⋅ 53 x 9 y 9
3 ⋅5 x y
39
22 x −6 y 2 ⋅ 26 x 6 y −12 22 ⋅ 26 x 6 x 8 y 2 y10
=
=
=
24 ⋅ 52 x −8 y −10
24 ⋅ 52 x 6 y12
28 x14 y12
24 x 8
= 2
24 ⋅ 52 x 6 y12
5
(6x y ) (36x y ) = ( 2 ⋅ 3x y ) ( 2 ⋅ 3 x y )
( 48x y )
( 2 ⋅ 3x y )
88.
4
−2
−4
−3
2
2
5
3
4
−4
−2
4
2
2
2
−3
2
26 ⋅ 36 y 8 y15
26 ⋅ 36 y 23
y 21
=
=
24 ⋅ 28 ⋅ 32 ⋅ 34 x16 x 9 x 4 y 2 212 ⋅ 36 x 29 y 2 26 x 29
5
3
2−4 ⋅ 3−4 x −16 y 8 ⋅ 26 ⋅ 36 x −9 y15
=
=
28 ⋅ 32 x 4 y 2
(x
89.
−2
) (x y z )
(x y z )
4
yz −3
5
−5 −1
6
−3 4
3
2
x −8 y 4 z −12 ⋅ x10 y −6 z 8 x10 y 4 y15 z 8 z 3 x10 y19 z11
y13
=
= 8 18 6 12 = 26 6 12 = 16
x18 y −15 z −3
xx y z
x y z
x z
( 24x y )(72x y ) = ( 2 ⋅ 3x y )( 2 ⋅ 3 x y ) = 2 ⋅ 3x
96x y ) (120x y ) ( 2 ⋅ 3x y ) ( 2 ⋅ 3⋅ 5x y ) 2 ⋅ 3x
90. (
−3
2
−5
3
2
3
−4
−3
3
5
−3
−5
2
3
3
2
3
23 ⋅ 23 ⋅ 3⋅ 32 x 2 x 5 y 2 y 3
25 ⋅ 33 x 7 y 5
3x
=
= 3 2
5
3
3 3 4 3
8
2 6 7
2 ⋅ 2 ⋅ 3⋅ 3⋅ 5x x y y
2 ⋅3 x y
2 y
2
3
−4
−3
3
5
−3
y 2 ⋅ 23 ⋅ 32 x 2 y −4
=
−5 3
y ⋅ 23 ⋅ 3⋅ 5x 3 y −3