This document reviews rules and properties of exponents and radicals. It contains examples of simplifying expressions using properties of exponents, evaluating radicals, combining radicals, rationalizing denominators, and changing between radical and exponential forms. The objective is to gain practice applying these rules and properties to simplify expressions involving exponents and radicals.

1. Exponents and Radicals
Objective: To review rules and
properties of exponents and
radicals.

2. Exponential Notation

3. Properties of Exponents

4. Properties of Exponents

5. Example 1
• Use the properties of exponents to simplify each
expression.
a) ( 3ab 4 )( 4ab 3 )

6. Example 1
• Use the properties of exponents to simplify each
expression.
a) ( 3ab 4 )( 4ab 3 )
( 3ab 4 )( 4ab 3 )
12 a 2b

7. Example 1
• Use the properties of exponents to simplify each
expression. You Try:
b)
2 3
(2 xy )

8. Example 1
• Use the properties of exponents to simplify each
expression. You Try:
b)
2 3
(2 xy )
(2 xy 2 )3
23 x 3 ( y 2 ) 3
8x3 y 6

9. Example 1
• Use the properties of exponents to simplify each
expression. You Try:
c)
3a ( 4a 2 ) 0

10. Example 1
• Use the properties of exponents to simplify each
expression. You Try:
c)
3a ( 4a 2 ) 0
3a( 4a 2 ) 0
3a(1) 3a, a
0

11. Example 1
• Use the properties of exponents to simplify each
expression. You Try:
d)
5x
y
3
2

12. Example 1
• Use the properties of exponents to simplify each
expression. You Try:
d)
5x
y
3
2
5x
y
3
2
52 ( x 3 ) 2
y2
25x 6
y2

13. Example 2
• Rewrite each expression with positive exponents.
a)
x
1

14. Example 2
• Rewrite each expression with positive exponents.
a)
x
1
x
1
1
x

15. Example 2
• Rewrite each expression with positive exponents.
1
b)
3x 2

16. Example 2
• Rewrite each expression with positive exponents.
1
b)
3x 2
1
3x 2
1 1
3 x 2
x2
3

17. Example 2
• Rewrite each expression with positive exponents.
• You Try:
12 a 3b 4
c)
4a 2 b

18. Example 2
• Rewrite each expression with positive exponents.
• You Try:
12 a 3b 4
c)
4a 2 b
12 a 3b 4
4a 2 b
12 a 3 a 2
4b 4b
3a 5
b5

19. Example 2
• Rewrite each expression with positive exponents.
• You Try:
d)
3x
y
2
2

20. Example 2
• Rewrite each expression with positive exponents.
• You Try:
d)
3x
y
3x
y
2
2
2
2
y
3x 2
2
y2
32 ( x 2 ) 2
y2
9x4

21. Radicals and Their Properties
• Definition of nth Root of a Number.
• Let a and b be real numbers and let n > 2 be a
positive integer. If
a = bn
then b is an nth root of a. If n = 2, the root is a
square root. If n = 3, the root is a cube root.

22. Radicals and Their Properties
• Principal nth Root of a Number.
• Let a be a real number that has at least one nth root.
The principal nth root of a is the nth root that has the
same sign as a. It is denoted by a radical symbol
n
a
• The positive integer n is the index of the radical, and
the number a is the radicand. If n = 2, omit the index
and write a .

23. Example 5
• Evaluate:
a)
36

24. Example 5
• Evaluate:
a)
36
36
6

25. Example 5
• Evaluate:
b)
36

26. Example 5
• Evaluate:
b)
36
36
6

27. Example 5
• Evaluate:
c)
36

28. Example 5
• Evaluate:
c)
36
36
DNE

29. Example 5
• Evaluate:
d)
3
125
64

30. Example 5
• Evaluate:
d)
3
125
64
3
125
64
3
125
3
64
5
4

31. Example 5
• Evaluate:
• You Try:
d)
3
27
8

32. Example 5
• Evaluate:
• You Try:
d)
3
27
8
3
27
8
3
27
3
8
3
2

33. Example 5
• Evaluate:
e)
5
32

34. Example 5
• Evaluate:
e)
5
32
5
32
2

35. Properties of Radicals

36. Properties of Radicals

37. Example 6
• Use the properties of radical to simplify each
expression.
a)
8
2

38. Example 6
• Use the properties of radical to simplify each
expression.
a)
8
8
2
2
16
4

39. Example 6
• Use the properties of radical to simplify each
expression.
b)
3
5
3

40. Example 6
• Use the properties of radical to simplify each
expression.
b)
3
3
5
5
3
3
1/ 3 3
5
51
5

41. Example 6
• Use the properties of radical to simplify each
expression.
c)
3
x
3

42. Example 6
• Use the properties of radical to simplify each
expression.
c)
3
x
x
3
3 1/ 3
x

43. Example 6
• Use the properties of radical to simplify each
expression.
d)
6
6
y6
y
6
| y|

44. Simplifying Radicals
•
An expression involving radicals is in simplest form
when the following conditions are satisfied.
1. All possible factors have been removed from the
radical.
2. All fractions have radical-free denominators
(accomplished by a process called rationalizing the
denominator).
3. The index of the radical is reduced.

45. Example 7
• Simplify each radical.
a)
32

46. Example 7
• Simplify each radical.
a)
32
32
16
2
4 2

47. Example 7
• Simplify each radical.
• You Try:
a)
24

48. Example 7
• Simplify each radical.
• You Try:
a)
24
24
4
6
2 6

49. Example 7
• Simplify each radical.
b)
4
48

50. Example 7
• Simplify each radical.
b)
4
4
48
48
4
16
4
3
24 3

51. Example 7
• Simplify each radical.
c)
75x3

52. Example 7
• Simplify each radical.
c)
75x3
75x3
25
3
x2
x
5x 3x

53. Example 7
• Simplify each radical.
• You Try:
c)
48x5

54. Example 7
• Simplify each radical.
• You Try:
c)
48x5
48x5
16
3
x4
x
4x2 3x

55. Example 8
• Simplify each radical.
a)
3
24

56. Example 8
• Simplify each radical.
a)
3
24
3
24
3
8
3
3
23 3

57. Example 8
• Simplify each radical.
b)
3
24a 4

58. Example 8
• Simplify each radical.
b)
3
24a 4
3
4
24a
3
8
3
3
3
a
3
3
a
2a3 3a

59. Example 8
• Simplify each radical.
• You Try:
c)
3
40x6

60. Example 8
• Simplify each radical.
• You Try:
c)
3
40x6
3
40x6
3
8
3
5
3
x6
2x 2 3 5

61. Example 9
• Combine each radical.
a)
2 48 3 27

62. Example 9
• Combine each radical.
a)
2 48 3 27
2 48 3 27
2
16
3 3
8
9
3 9
3
3
3

63. Example 9
• Combine each radical.
• You Try:
b)
3
16x
3
54x
4

64. Example 9
• Combine each radical.
• You Try:
b)
3
3
16x
3
16x
3
54x
4
54x
4
3
8
2
3
3
2x
3
27x
2 x 3x
3
3
2x
3
2x
( 2 3 x )3 2 x

65. Example 10
• Rationalize the denominator of each expression.
a)
5
3

66. Example 10
• Rationalize the denominator of each expression.
a)
5
3
5
3
3
3
5 3
3

67. You Try
• Rationalize the denominator of each expression.
• You Try:
b)
1
2

68. Example 10
• Rationalize the denominator of each expression.
• You Try:
b)
1
2
1
2
2
2
2
2

69. Example 11
• Rationalize the denominator of each expression.
2
a)
3
7
2
(3
(3
7 ) (3
7)
7)
2(3 7 )
9 7
3
7

70. You Try
• Rationalize the denominator of each expression.
• You Try:
b)
3
4
5

71. You Try
• Rationalize the denominator of each expression.
• You Try:
b)
3
4
5
3
(4
(4
5 ) (4
5)
5)
12 3 5
11

72. Rational Exponents
• The numerator of a rational exponent denotes the
power to which the base is raised, and the
denominator denotes the index or the root to be
taken.

73. Example 13
• Change the base from radical to exponential form.
a)
3

74. Example 13
• Change the base from radical to exponential form.
a)
3
3
1/ 2
3

75. Example 13
• Change the base from radical to exponential form.
b)
x
5

76. Example 13
• Change the base from radical to exponential form.
b)
x
5
x
5
5 1/ 2
(x )
x
5/ 2

77. You Try
• Change the base from radical to exponential form.
• You Try:
c)
3
y
4

78. You Try
• Change the base from radical to exponential form.
• You Try:
c)
3
3
y
4
y
4
4 1/ 3
(y )
y
4/3

79. Example 14
• Change the base from exponential to radical form.
a) ( x y ) 3 / 2

80. Example 14
• Change the base from exponential to radical form.
a) ( x y ) 3 / 2
( x y)3 / 2
( x y)3

81. You Try
• Change the base from exponential to radical form.
b)
x 3 / 4 y1/ 4

82. You Try
• Change the base from exponential to radical form.
b)
x 3 / 4 y1/ 4
x3 / 4 y1/ 4
( x3 y)1/ 4
4
x3 y

83. Example 15
• Simplify each rational expression.
a)
( 32 ) 3 / 5

84. Example 15
• Simplify each rational expression.
a)
( 32 ) 3 / 5
( 32 )
3/ 5
1/ 5 3
(( 32 ) )
( 2)
3
8

85. Example 15
• Simplify each rational expression.
• You Try:
b)
(27 )
2/3

86. Example 15
• Simplify each rational expression.
• You Try:
b)
(27 )
2/3
(27)
2/3
1/ 3
((27) )
2
(3)
2
1
9

87. You Try
• Simplify each rational expression.
• You Try:
c)
(64 ) 2 / 3

88. You Try
• Simplify each rational expression.
• You Try:
c)
(64 ) 2 / 3
(64 )
2/3
1/ 3 2
(( 64 ) )
4
2
16

89. You Try
• Simplify each rational expression.
• You Try:
d) (16 )
3/ 4

90. You Try
• Simplify each rational expression.
• You Try:
d) (16 )
(16)
3/ 4
3/ 4
1/ 4
((16) )
3
2
3
1
8