1. Rules for exponents are reviewed, including that when multiplying or dividing terms with the same base, the exponents are added or subtracted. 2. Several examples are worked through applying these rules to simplify expressions involving exponents. 3. Key points about negative exponents and operations involving them are explained.

1. 1. ʿ F ʾ 1
F ˈ 3 F F
ʿ F
ʿ F
F
ʿ F
2. ʿ ˈ ʿ F
2
3. ʿ F F F
3.1 F
3.2
3.3 ʿ F F F
3.4 ʿ F F F F
3.5
3.6
3.7
3.8 F F ʿ

2. F 2
F × F
1. F a ≠ 0 F F F
. a 0
= 1 . a m
× a n
= a nm+
. a 1
= 0 . a n
= n
a−
1
2. m 599,1
× m F F
. m 600,1
. m 599,1
. 2 m 599,1
. 2 m 600,1
3. - 







2
3
39
13
a
a
F F
. - a
2
1
. - a
3
1
. - a
3
1 5
. - a
3
1 6
4. (2 × 105
) × (3 × 106
) F F
. 5 × 1011
. 5 × 1030
. 6 × 1030
. 6 × 1011
5. (- x 2
) ( 2 x 3
) (-3 x 4
) F F
. 123 x 9
. 23 x 9
. 6 x 9
. 9 x 9
6. (-955
) × (-9125
) F F
. 81180
. 81 180−
. - 9180
. 9180

3. 7. 3
0
a
a
F F
. 0 . 1
. a . 3
1
a
8. ( 5 a 2
b )0
F F
. 0 . 1
. 2 . 5
9. 5
43
3
33 ×
F F
. 3 . 32
. 33
. 312
10. F F F
. (- 2a 2
b ) 2
= (-2 )2
a 4
b 2
. 2 (a 2
b ) 2−
= - 2 a 4−
b 2−
. (-a 2
)3
= (-1)3
( a 2
)3
. (- x 2
y 3
)3
= x 4
y 6
*******************

4. F 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

5. ʿ F 2
F F
1. F
2. ˈ
F F F ˆ F

6. ˈ ˈ
ˈ
F a ˈ F ˈ 0 m n ˈ
a m
× a n
= a nm+
F 1 F 27
× 28
27
× 28
= 2 87+
= 215
F 2 F ( m 4
× m 5
) × (m 8
× m 2
)
(m 4
× m 5
) × (m 8
× m 2
) = (m 54+
) × (m 28+
)
= m 9
× m 10
= m 109+
= m 19
F 3 F a 2
b 3
× a 3
b
a 2
b 3
× a 3
b = a 2
× b 3
× a 3
× b
= (a 2
× a 3
)×(b 3
× b )
= a 32+
× b 13+
= a 5
b 4

7. F a ˈ m n ˈ
a m
× a n
= a nm+

8. ʿ F
2.1
F F
F
F 107
= 104
103
106
102
10 10
10
2
10
5
10
10
10
6
10
20
10
15
10
9

9. ʿ F
2.1
F F
F
106
102
10 10
10
2
10
5
10
10
10
6
10
20
10
15
10
9
10
12
10
3
10
4
10
8
10
6

10. ʿ F
2.2
F
F a 2
× a 5
= a 52+
= a 7
1. 53
× 59
= .= ..
2. 24
× 23
× 24
= = ..= ..
3. m 10
× m 2
× m 3
= ..= ..
4. ( b 4
× b 3
)×( b 7
× b 5
) = = .= ...
5. a 2
× b 3
× a 4
× b 5
= .= ..=

11. ʿ F
2.2
1. 53
× 59
= 5 93+
= 512
2. 24
× 23
× 24
= 24
×23
× 24
= 2 434 ++
= 211
3. m 10
× m 2
× m 3
= m 3210 ++
= m 15
4. (b 4
× b 3
)×(b 7
× b 5
) = b 34+
× b 57+
= b 127+
= b 19
5. a 2
× b 3
× a 4
× b 5
= (a 2
× a 4
)(b 3
× b 5
) = a 42+
b 53+
= a 6
b 8

12. ˈ ˈ F
F a b ˈ a ≠ 0, b ≠ 0 n ˈ
(a b )n
= a n
b n
F 1 F (7 × 5)3
(7 × 5)3
= 73
× 53
= 343 × 125
= 42,875
F 2 F ( 3a b )2
( 3a b )2
= 32
a 2
b 2
= 9 a 2
b 2
F 3 F F ( 2 x y )2
× (x y )5
( 2 x y )2
× (x y )5
= ( 22
× x 2
× y 2
) ( x 5
× y 5
)
= 22
×( x 2
× x 5
) ( y 2
× y 5
)
= 22
× x 52+
× y 52+
= 22
x 7
y 7
= 4 x 7
y 7

13. F a b ˈ n ˈ
(a b )n
= a n
b n

14. ʿ F
2.3
F F
F ( 6× 8 )2
= 62
× 82
= 36 × 64
= 2,304
1. ( 12 × 9 )2
=
=
=
2. ( 25 × 16 )3
=
=
=
3. ( 5a b )2
=
=
= ...........................
4. ( 12 x y )4
( x y )3
=
=
= ...........................
5. (a 2
b 3
) (a b )5
=
=
= ...............................................

15. ʿ F
2.3
1. ( 12 × 9 )2
= 122
× 92
= 144 × 81
= 11,664
2. ( 25 × 16 )3
= 253
× 163
= 15,625 × 4,096
= 64,000,000
3. ( 5ab )2
= 52
a 2
b 2
= 25 a 2
b 2
4. ( 12 xy )4
( xy )3
= (12 4
x 4
y 4
) ( x 3
y 3
)
= 12 4
x 34+
y 34+
= 20,736 x 7
y 7
5. (a 2
b 3
) (ab )5
= (a 2
b 3
)(a 5
b 5
)
= a 52+
b 53+
= a 7
b 8

16. ˈ
ˈ F
F a ˈ m n ˈ
(a m
)n
= a mn
1. ˈ
F 1 F ( 32
)3
( 32
)3
= 3 32×
= 36
= 729
F 2 F {( 23
)2
}2
{( 23
)2
}2
= ( 23
) 22×
= ( 23
)4
= 2 43×
= 212
= 4,096
F 2 F ( 22
)4
+ ( 33
)2
( 22
)4
+ ( 33
)2
= 2 42×
+ 3 23×
= 28
+ 36
= 256 + 729
= 985

17. F a ˈ m n ˈ
(a m
)n
= a mn

18. ʿ F
2.4
F F
F ( 92
)3
= 9 32×
= 96
= 531,441
1. ( x 7
)3
= ..
= .
= .
2. ( a 2
b ) 3
= ..
= .
= .
= .
3.
32
3
2






= ..
= .
= .
= .............................

19. 4. {(0.3)2
}3
= ..
= .
= .
= .............................
5. {(23
)2
}2
= ..
= .
= .
= .............................
6. {(a 4
)3
}2
= ..
= .
= .
= .............................
7. (a 2
)3
× (a 3
)5
= ..
= .
= .
= .............................

20. ʿ F
2.4
1. (x 7
)3
= x 37×
= x 21
2. (a 2
b ) 3
= a 32×
× b 31×
= a 6
b 3
3.
32
3
2








= 31
32
3
2
×
×
= 3
6
3
2
=
27
64
= 2
27
10
4. {(0.3)2
}3
= ( 0.3 ) 32×
= ( 0.3 )6
= 0.000729

21. 5. {(23
)2
}2
= (23
) 22×
= (23
)4
= 2 43×
= 212
= 4,096
6. {(a 4
)3
}2
= (a 4
) 23×
= (a 4
)6
= a 64×
= a 24
7. (a 2
)3
× (a 3
)5
= a 32×
× a 53×
= a 6
× a 15
= a 156+
= a 21

22. ˈ
ˈ F
F a ˈ m n ˈ
( a m
)n
= nm
a ×
= a mn
ˈ
F 1 F (-23
)2
(-23
)2
= {(-1)(23
)}2
= (-1 )2
× 2 23×
= 1×26
= 26
F 2 F {(-a )4
}5
{(-a )4
}5
= (-a ) 54×
= (-a )20
= (-1)20
× 20
a
= 20
a

23. ʿ F
2.5
F F
F (- 22
)4
= {(-1) × (22
)}4
= (-1)4
×2 42×
= 1×28
= 28
1. (-32
)3
= .
= .
= .
2. (- x 3
)4
= .
= .
= .................................
3. (- 52
)2x = .
= .
= .................................
4. {(-a) 3
}4
= .
= .
= .................................
= ..
= ..

24. ʿ F
2.5
1. (-32
)3
= {(-1) ×32
}3
= (-1)3
×3 32×
= (-1) 3
×36
= (-1)×36
= - 36
2. (- x 3
)4
= {(-1) x 3
}4
= (-1)4
x 43×
= x 12
3. (- 52
)2x = {(-2)x 2
}5
= (-2 )5
x 52×
= -32 10
x
4. {(-a) 3
}4
= (-a) 43×
= (-a)12
= {(-1) (a)}12
= (-1 ) 12 12
a
= 12
a

25. F F ʿ F 2
F F F
1. 216
F F F
. 28
× 28
. 24
× 24
. 22
× 28
. 218
× 22
2. (D3
× D2
)× (D4
× D6
) F F F
. D4
. D15
. D30
. D 4−
3. (3× 5) 3
F
. 135 . 375
. 225 . 492
4. [(0.3) 2
]3
F
. 0.729 . 0.0729
. 0.00729 . 0.000729
5. (-2x2
)5
F F F
. 32x7
. - 32x7
. -32x10
. 32x10
6.
27
64
F F
.
3
2
3
2








.
3
3
8






.
2
3
8






.
6
3
2






7. 8× 22
× 24
F F F
. 29
. 28
. 27
. 26

26. 8. 27,000 F F
. (3× 9)3
. (3× 3× 3)3
. (3× 9)2
. (2× 3× 5)3
9. 52
× (-5)3
F
. 55
. (-5)1
. (-5)5
. 5
10. F F 27 243 3
F F F
( )
. 36
. 37
. 38
. 39
************************

27. F ʿ F 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

28. F F
F
F a ˈ F F F m n ˈ
m > n F a m
÷ a n
= a nm−
m ˈ n ˈ
m > n a ≠ 0 F F
n
m
a
a
= a nm−
F 1 3
4
5
5
3
4
5
5
= 5 34−
= 5
F 2 4
6
)4(
)4(
−
−
4
6
)4(
)4(
−
−
= 46
)4( −
−
= 2
)4(−

29. m ˈ n ˈ
m > n a ≠ 0 F F n
m
a
a
= a nm−

30. ʿ F
2.6
F ( )
F 4
9
3
3
= 3 49−
= 35
1. 5
43
2
22 ×
=
=
=
2. 14
20
9
9
=
=
3. 7
20
)(
)(
a
a
−
−
=
=
=
4. 2
8
2
4
a
a
=
=
=

31. 5.
m
m
5
125 10
=
=
=
6. 4
6
2
1
2
1












= .
= .
= .

32. ʿ F
2.6
1. 5
43
2
22 ×
= 5
43
2
2 +
= 5
7
2
2
= 2 57−
= 22
2. 14
20
9
9
= 9 1420−
= 96
3. 7
20
)(
)(
a
a
−
−
= 720
)( −
−a
= 13
)( a−
4. 2
8
2
4
a
a
= 2
8
2
a
a
= 2(a 28−
)
= 2a 6

33. 5.
m
m
5
125 10
=
m
m10
25
= 25 (m 110−
)
= 25 9
m
6. 4
6
2
1
2
1












=
46
2
1 −






=
2
2
1







34. F F
F F 6
3
2
2
6
3
2
2
=
222222
222
×××××
××
=
222
1
××
6
3
2
2
= 3
2
1
1
n
m
a
a
= nm
a −
6
3
2
2
= 2 63−
= 2 3−
2
1 2 F 3
2
1
= 2 3−
F n = 3 , a = 2
n
a
1
= a n−
a n
= n
a−
1
a F F F n ˈ
a n−
= n
a
1

35. m ˈ n ˈ
F m < n a ≠ 0 F F n
m
a
a
= mn
a −
1

36. ʿ F
2.7
F ( )
F 18
10
a
a
= a 1810−
= a 8−
= 8
1
a
1. 7
3
5
5
=
=
=
2. 4
2
)3.0(
)3.0(
−
=
=
=
3. 6
4
2
)2(−
=
=
=

37. 4. 5
3
2
6
k
k
=
=
=
5. 54
32
18
27
nm
nm
=
=
=
6. 5
3
3
1
3
1












=
=
=

38. ʿ F
2.7
1. 7
3
5
5
= 5 73−
= 5 4−
= 4
5
1
2. 4
2
)3.0(
)3.0(
−
= 4
2
)3.0(
)3.0(
= (0.3) 42−
= (0.3) 2−
= 2
)3.0(
1
3. 6
4
2
)2(−
= (-2) 64−
= (-2) 2−
= 2
)2(
1
−
4. 5
3
2
6
k
k
= 5
3
3
k
k
= 3(k 53−
)
= 3 k 2−
= 2
3
k

39. 5. 54
32
18
27
nm
nm
= 54
32
2
3
nm
nm
=
2
3 5342 −−
nm
=
2
3 22 −−
nm
= 22
2
3
nm
6. 5
3
3
1
3
1












=
53
3
1
−






=
2
3
1
−






= 2
3
1
1






= 1× 32
= 32

40. F
a F F F
a 0
= 1
F F 1) 3
3
a
a
2) 5
5
9
9
1) 3
3
a
a
1 3
3
a
a
=
aaa
aaa
××
××
= 1
2 3
3
a
a
= 33−
a
= 0
a
= 1
2) 5
5
9
9
1 5
5
9
9
=
99999
99999
××××
××××
= 1
2 5
5
9
9
= 55
9 −
= 90
= 1

41. m ˈ n ˈ
m = n a ≠ 0 F F
n
m
a
a
= nm
a −
= 0
a
= 1
0
a = 1

42. ʿ F
2.8
F
F 3
3
5
5
= 5 33−
= 50
= 1
1. 7
43
2
22 ×
= .. .
= .. .
= .. .
2. 5
53
2
14
mn
nm
−
−
= .. .
= .. .
= .. .
3. n
nn
mm
mm
30
2
×
×
= .. .
= .. .
= .. .

43. 4. 7
44
33
)3(3
×
−×
= .. .
= .. .
= .. .
5. 0
55
x
xx ×−
= .. .
= .. .
= .. .

44. ʿ F
2.8
1. 7
43
2
22 ×
= 7
43
2
2 +
= 7
7
2
2
= 77
2 −
= 0
2
= 1
2. 5
53
2
14
mn
nm
−
−
= 5513
7 −−
nm
= 02
7 nm
= 2
7m
3. n
nn
mm
mm
30
2
×
×
= n
nn
m
m
30
2
+
+
= n
n
m
m
3
3
= nn
m 33 −
= 0
m
= 1

45. 4. 7
44
33
)3(3
×
−×
= 7
44
33
33
×
×
= 71
44
3
3
+
+
= 8
8
3
3
= 88
3 −
= 0
3
= 1
5. 0
55
x
xx ×−
= 5)5( +−
x
= 0
x
= 1

46. ʿ F
2.9
F F F F F F
F F F F F F
F 1. 7 × 32
1. 54
2. 33 × 24
3. 22
×32
6 3
2 6
5 2 8
1. 8 × 23
× (-2 )4
2. ( 42
× 43
) ÷ 44
3. 3
8
7
7
5. ( 5 × 3 )2
7. 27
÷ 23
9. 3
24
2
43 ×
11. ( 52
× 52
)
1. 2
8
5
5
2. ( 7 × 5 )3
4. ( 32
)3
6. 2 ×23
×24
8. 32
× 93
10. ( 6 × 8 )2
12. 22
× 32
1 3
2
1
3 8 4
5
12
7 9 11
6 2 10

47. ʿ F
2.9
F F F F F F
F F F F F F
0 2 4 2
5 5 2 3
6 1 6 8 0 7
2 2 5 7 4 2
5 3 6 5 9
1 6 1 6 2 5
3 8 4
5
12
7 9 11
1 26 101

48. 1. F a n ˈ F
n
a = a × a × a × .× a
n
2. a F F F n
n
n
a
a
1
=−
3. a F F F
10
=a
F a b ˈ a ≠ 0, b ≠ 0
m n ˈ F
1. nmnm
aaa +
=×
2. (ab)n
= nn
ba
3. mnnm
aa =)(
4. nm
n
m
a
a
a −
=

49. ʿ F
2.10
F F F F F F
( 0≠a )
F 0
a = 1 1
a = 0
( F ) ( F F )
20
= 1 21
= 0
( ) ( )
a n
=
n
1 n
a−
= n
a
1
( ) ( )
2
a ×
5
a = 7
a ( 7
a )3
= 10
a
( ) ( )
4
a ÷ 5
a = 1−
a 5
a ÷ 4
a = 1−
a
( ) ( )

50. ʿ F
2.10
F F F F F F
( 0≠a )
20
= 1 21
= 0
( F ) ( F F )
a n
=
n
1 n
a−
= n
a
1
( F F ) ( F )
2
a ×
5
a = 7
a ( 7
a )3
= 10
a
( F ) ( F F )
4
a ÷ 5
a = 1−
a 5
a ÷ 4
a = 1−
a
( F ) ( F F )

51. F F ʿ F 3
F F F
1. 37
÷ 32
F F F
. 39
. 37
. 35
. 33
2. 3
42
5
55 ×
F F F
. 53
. 52
. 51
. 50
3. 2
7
13
143
a
a
F F F
. 13a 5
. 11a 5
. 13a 9
. 11a 9
4. 4
2
)4.0(
)4.0(
−
F F F
.
)4.0(
1
. 3
)4.0(
1
. 6
)4.0(
1
. 2
)4.0(
1
5. 22
3
4
yx
F F F
. 22
9
12
yx
xy
. 54
32
21
28
yx
yx
. 54
22
9
12
yx
yx
. 45
22
21
28
yx
yx
6. n
nn
aa
aa
30
2
×
×
F F F
. 4 . 3
. 2 . 1

52. 7. 5
.3
5
1
5
1












F F F
. 52
. ( )
5
1
2−
.
2
5
1






.
( )2
5
1
8. 0
22
3
33 ×−
F F F
. 32
. 1
. 34
. 3
9. 3
62
7
77
−
−
×
F F F
. 75
. 77
. 79
. 711
10. 4x
= 12
4
1
, x F F
. 4 .
12
1
. 12 . 12
**************************

53. F F ʿ F 3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

54. F F ʿ F 4
F × F
1. 3
0
a
a
F F
. 0 . 1
. a . 3
1
a
2. - 





2
3
39
13
a
a
F F
. - a
2
1
. - a
3
1
. - 5
3
1
a . - 6
3
1
a
3. 5
43
3
33 ×
F F
. 3 2
. 3
. 3
3 . 12
3
4. F 0≠a F F F
. 10
=a . nmnm
aaa +
=×
. 01
=a . n
n
a
a
1
=−
5. (2 × 105
) × (3 × 106
) F F
. 5 × 1011
. 6 × 1011
. 6 × 1030
. 5 × 1030
6. (-955
)×(-9125
) F F
. 81180
. 81 180−
. -9180
. 9180
7. (5 ba2
)0
F F
. 0 . 1
. 2 . 5

55. 8. mm ×599,1
F F
. 599,1
m . 600,1
m
. 599,1
2m . 600,1
2m
9. (- x 2
) (2 x 3
) (-3 x 4
) F F
. 123 x 9
. 23 x 9
. 6 x 9
. 9 x 9
10. F F F
. ( 24222
)2()2 baba −=− . -2 ( 2422
2) −−−
−= baba
. ( 32332
)()1() aa −=− . (- 64332
) yxyx =

56. F F ʿ F 4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.