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dug84356_ch09a.qxd 9/14/10 2:11 PM Page 580

580

9-24

Chapter 9 Radicals and Rational Exponents

Solution

U Calculator Close-Up V

Check that

͙8

ෆ ϩ ͙18

ෆ ϭ 5͙2

ෆ.

a) ͙8ෆ ϩ ͙18

ෆ ϭ ͙4

ෆ и ͙2

ෆ ϩ ͙9

ෆ и ͙2

ϭ 2͙2

ෆ ϩ 3͙2

ϭ 5͙2

Note that ͙8ෆ ϩ ͙18

b)

͙26

ෆ.

͙2x

ෆ3 Ϫ ͙4x

ෆ2 ϩ 5͙18x

ෆ3 ϭ ͙xෆ2 и ͙2x

ෆ Ϫ 2x ϩ 5 и ͙9x

ෆ2 и ͙2x

ϭ x͙2x

ෆ Ϫ 2x ϩ 15x͙2x

ϭ 16x͙2x

ෆ Ϫ 2x

c)

͙ෆ

16x4y3 Ϫ ͙ෆ

54x4y3 ϭ ͙ෆ

8x3y3 и ͙2x

ෆ Ϫ ͙ෆ

27x3y3 и ͙2x

3

3

3

3

3

ϭ 2xy͙2x

ෆ Ϫ 3xy͙2x

3

3

3

ϭ Ϫxy͙2x

3

Now do Exercises 13–28

n

n

n

The product rule for radicals, ͙aෆ и ͙

bෆ ϭ ͙

ෆ, allows multiplication of radicals with

ab

the same index, such as

ෆ ϭ ͙15

ෆ,

͙5ෆ и ͙3

3

3

3

͙

2ෆ и ͙

5ෆ ϭ ͙

ෆ,

10

and

5

5

5

͙

xෆ2 и ͙

xෆ ϭ ͙

xෆ3.

CAUTION The product rule does not allow multiplication of radicals that have dif3

5ෆ.

ferent indices. We cannot use the product rule to multiply ͙2ෆ and ͙

E X A M P L E

3

Multiplying radicals with the same index

Multiply and simplify the following expressions. Assume the variables represent positive

numbers.

Students often write

a) 5͙6ෆ и 4͙3

b)

͙3a

ෆ2 и ͙6a

3

3

4ෆ и ͙

4ෆ

c) ͙

d)

Ί๶ Ί๶

4

x2

ᎏᎏ

8

a) 5͙6ෆ и 4͙3

ෆ ϭ 5 и 4 и ͙6

ෆ и ͙3

ϭ 20͙18

ϭ 20 и 3͙2

ෆ ͙18

ෆ ϭ ͙9

ෆ и ͙2

ෆ ϭ 3͙2

Although this is correct, you should

get used to the idea that

Because of the definition of a square

root, ͙a

ෆ и ͙a

ෆ ϭ a for any positive

number a.

x3

ᎏᎏ и

2

Solution

͙15

ෆ и ͙15

ෆ ϭ ͙225

ෆ ϭ 15.

͙15

ෆ и ͙15

ෆ ϭ 15.

4

ϭ 60͙2

b)

͙3a

ෆ2 и ͙6a

ෆ ϭ ͙18a

ෆ3

ϭ ͙ෆ

9a и ͙2a

2

ϭ 3a͙2a

Simplify.

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9-25

9.3

c)

581

3

3

3

͙

4ෆ и ͙4ෆ ϭ ͙16

ϭ ͙8ෆ и ͙2ෆ

3

3

Simplify.

ϭ 2͙2ෆ

3

d)

Ίxᎏ๶2ᎏ и Ίxᎏ๶8ᎏ ϭ Ίᎏ๶1xᎏ6

3

4

2

5

4

4

͙ෆ

x4 и ͙xෆ

ϭ ᎏᎏ

4

͙16

4

4

Product and quotient rules for radicals

4

xෆ

ϭ ᎏᎏ

2

Simplify.

Now do Exercises 29-42

We find a product such as 3͙2ෆ(4͙2ෆ Ϫ ͙3

ෆ ) by using the distributive property as we do when multiplying a monomial and a binomial. A product such as

(2͙3ෆ ϩ ͙5ෆ )(3͙3ෆ Ϫ 2͙5ෆ ) can be found by using FOIL as we do for the product

of two binomials.

E X A M P L E

4

Multiply and simplify.

ෆ)

a) 3͙2ෆ (4͙2ෆ Ϫ ͙3

2

b) ͙aෆ (͙aෆ Ϫ ͙aෆ

)

c) (2͙3

ෆ ϩ ͙5

ෆ )(3͙3ෆ Ϫ 2͙5

ෆ)

d) (3 ϩ ͙x

Ϫ 9 )2

3

3

3

Solution

a) 3͙2ෆ (4͙2ෆ Ϫ ͙3

ෆ ) ϭ 3͙2

ෆ и 4͙2

ෆ Ϫ 3͙2

ෆ и ͙3

ϭ 12 и 2 Ϫ 3͙6

Distributive property

ෆϭ2

Because ͙2ෆ и ͙2

ෆ и ͙3

ෆ ϭ ͙6

and ͙2

ϭ 24 Ϫ 3͙6

b)

3

3

͙aෆ (͙aෆ Ϫ ͙aෆ2 ) ϭ ͙

aෆ2 Ϫ ͙aෆ3

3

3

3

Distributive property

ϭ ͙aෆ Ϫ a

3

2

ෆ ϩ ͙5

ෆ )(3͙3ෆ Ϫ 2͙5

ෆ)

c) (2͙3

F

O

I

L

Ά

Ά

Ά

Ά

ϭ 2͙3

ෆ и 3͙3

ෆ Ϫ 2͙3

ෆ и 2͙5

ෆ ϩ ͙5

ෆ и 3͙3

ෆ Ϫ ͙5

ෆ и 2͙5

ϭ 18 Ϫ 4͙15

ෆ ϩ 3͙15

ෆ Ϫ 10

ϭ 8 Ϫ ͙15

d) To square a sum, we use (a ϩ b)2 ϭ a 2 ϩ 2ab ϩ b 2:

(3 ϩ͙xෆ

Ϫ 9 ϩ (͙ ෆ

x Ϫ 9 )2

Ϫ 9)2 ϭ 32 ϩ 2 и 3͙x

ϭ 9 ϩ 6͙xෆ

Ϫ9 ϩxϪ9

ϭ x ϩ 6͙x

Ϫ9

Now do Exercises 43-56

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Chapter 9 Radicals and Rational Exponents

CAUTION We can’t simplify ͙xෆ

Ϫb

Ϫ 9 in Example 4(d), because in general ͙aෆ

͙a

ෆ Ϫ ͙b

ෆ. For example, ͙ෆ6

25 Ϫ 1ෆ ϭ ͙9

ෆ ϭ 3 and ͙25

ෆ Ϫ ͙16

ෆ ϭ 1.

ෆ ϩ ͙b

ෆ.

Find an example where ͙a

ϩ b ͙a

U3V Conjugates

Recall the special product rule (a ϩ b)(a Ϫ b) ϭ a 2 Ϫ b 2. The product of the sum

4 ϩ ͙3

ෆ and the difference 4 Ϫ ͙3

ෆ can be found by using this rule:

(4 ϩ ͙3ෆ)(4 Ϫ ͙3ෆ) ϭ 42 Ϫ (͙3ෆ)2 ϭ 16 Ϫ 3 ϭ 13

The product of the irrational number 4 ϩ ͙3

ෆ and the irrational number 4 Ϫ ͙3

ෆ is the

ෆ and 4 Ϫ ͙3ෆ are called

rational number 13. For this reason the expressions 4 ϩ ͙3

conjugates of one another. We will use conjugates in Section 9.4 to rationalize some

denominators.

E X A M P L E

5

Multiplying conjugates

Find the products. Assume the variables represent positive real numbers.

a) (2 ϩ 3͙5ෆ )(2 Ϫ 3͙5ෆ )

b) (͙3ෆ Ϫ ͙2

ෆ )(͙3ෆ ϩ ͙2

ෆ)

c) (͙2x

ෆ Ϫ ͙yෆ )(͙2x

ෆ ϩ ͙yෆ )

Solution

a) (2 ϩ 3͙5ෆ )(2 Ϫ 3͙5ෆ ) ϭ 22 Ϫ (3͙5ෆ )2

ϭ 4 Ϫ 45

(a ϩ b)(a Ϫ b) ϭ a 2 Ϫ b 2

(3͙5ෆ )2 ϭ 9 и 5 ϭ 45

ϭ Ϫ41

b) (͙3ෆ Ϫ ͙2

ෆ )(͙3ෆ ϩ ͙2

ෆ) ϭ 3 Ϫ 2

ϭ1

ෆ Ϫ ͙yෆ )(͙2x

ෆ ϩ ͙yෆ ) ϭ 2x Ϫ y

c) (͙2x

Now do Exercises 57–66

U4V Multiplying Radicals with Different Indices

The product rule for radicals applies only to radicals with the same index. To multiply

with rational exponents. If the exponential expressions have the same base, apply the

product rule for exponents (am и an ϭ amϩn) to get a single exponential expression and

then convert back to a radical [Example 6(a)]. If the bases of the exponential

expression are different, get a common denominator for the rational exponents,

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9-27

9.3

583

convert back to radicals and then apply the product rule for radicals (͙aෆ и ͙bෆ ϭ

n

ෆ ) to get a single radical expression [Example 6(b)].

͙ab

n

E X A M P L E

6

n

Write each product as a single radical expression.

a) ͙ෆ2 и ͙ෆ2

3

3

4

a) ͙

2ෆ и ͙

2ෆ ϭ 21͞3 и 21͞4

Check that

ϭ2

12

3

ϭ ͙2ෆ

12

Write in exponential notation.

Product rule for exponents: 1ᎏᎏ ϩ 1ᎏᎏ ϭ ᎏ7ᎏ

7͞12

͙2

ෆ и ͙2

ෆ ϭ ͙128

ෆ.

4

3

Solution

U Calculator Close-Up V

3

b) ͙ෆ2 и ͙3

4

7

4

12

ϭ ͙128

12

3

b) ͙

ෆ2 и ͙3

ෆ ϭ 21͞3 и 31͞2

ϭ 22͞6 и 33͞6

Write in exponential notation.

Write the exponents with the LCD of 6.

ϭ ͙ෆ

22 и ͙ෆ

6

6

ϭ ͙ෆ

22 и 33

ϭ ͙108

22 и 33 ϭ 4 и 27 ϭ 108

6

6

Now do Exercises 67–74

CAUTION Because the bases in 21͞3 и 21͞4 are identical, we can add the exponents

[Example 6(a)]. Because the bases in 22͞6 и 33͞6 are not the same, we

as a sixth root and use the product rule for radicals.

Warm-Ups

Fill in the blank.

1.

radicals have the same index and the same

2. The

property is used to combine like

with the same

.

4. The

of 2 Ϫ ͙3

ෆ is 2 ϩ ͙3

ෆ.

True or false?

5. ͙3

ෆ ϩ ͙3

ෆ ϭ ͙6

6. ͙8

ෆ ϩ ͙2

ෆ ϭ 3͙2

7. 2͙3

ෆ и 3͙3

ෆ ϭ 6͙3

8. 2͙5

ෆ и 3͙2

ෆ ϭ 6͙10

3

3

9. ͙2ෆ и ͙2ෆ ϭ 2

ෆ(͙3ෆ Ϫ ͙2

ෆ ) ϭ ͙6

ෆϪ2

10. ͙2

ෆ )2 ϭ 2 ϩ 3

11. (͙2ෆ ϩ ͙3

ෆ )(͙3ෆ ϩ ͙2

ෆ) ϭ 1

12. (͙3ෆ Ϫ ͙2

9.3

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10:45 AM

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Exercises

584

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Chapter 9 Radicals and Rational Exponents

U Study Tips V

• If you must miss class, let your instructor know. Be sure to get notes from a reliable classmate.

• Take good notes in class for yourself and your classmates.You never know when a classmate will ask to see your notes.

All variables in the following exercises represent positive

numbers. Simplify the sums and differences. Give exact

Simplify the products. Give exact answers. See Examples 3

and 4.

1. ͙3

ෆ Ϫ 2͙3

2. ͙5

ෆ Ϫ 3͙5

3. 5͙7x

ෆ ϩ 4͙7x

4. 3͙6a

ෆ ϩ 7͙6a

5. 2͙2ෆ ϩ 3͙2ෆ

6. ͙4ෆ ϩ 4͙4ෆ

3

3

3

3

7. ͙3

ෆ Ϫ ͙5

ෆ ϩ 3͙3

ෆ Ϫ ͙5

8. ͙2

ෆ Ϫ 5͙3

ෆ Ϫ 7͙2

ෆ ϩ 9͙3

9. ͙2ෆ ϩ ͙xෆ Ϫ ͙2ෆ ϩ 4͙xෆ

3

3

3

3

3

ෆ и 3͙10

31. 2͙5

32. (3͙2ෆ)(Ϫ4͙10

ෆ)

33. 2͙7a

ෆ и 3͙2a

34. 2͙5c

ෆ и 5͙5

35. ͙9ෆ и ͙27

36. ͙5ෆ и ͙100

37. (2͙3ෆ )2

38. (Ϫ4͙2

ෆ )2

4

4

39.

͙ෆ

5x3 и ͙ෆ

8x4

40.

͙ෆ

3b3 и ͙ෆ

6b5

41.

Ίxᎏ๶3ᎏ и Ίᎏ2๶xᎏ7

42.

Ίaᎏ๶2ᎏ и Ίaᎏ๶4ᎏ

3

11. ͙xෆ Ϫ ͙2x

ෆ ϩ ͙xෆ

3

ෆ и ͙7

30. ͙5

3

3

10. ͙5y

ෆ Ϫ 4͙5y

ෆ ϩ ͙xෆ ϩ ͙xෆ

3

29. ͙3

ෆ и ͙5

3

12. ͙ab

ෆ ϩ ͙aෆ ϩ 5͙aෆ ϩ ͙ab

3

3

Simplify each expression. Give exact answers. See Example 2.

ෆ ϩ ͙28

13. ͙8

4

3

5

4

4

3

2

3

43. 2͙3

ෆ(͙6ෆ ϩ 3͙3

ෆ)

ෆ(͙3ෆ ϩ 3͙5

ෆ)

44. 2͙5

ෆ ϩ ͙24

14. ͙12

ෆ(͙10

ෆ Ϫ 2)

45. ͙5

ෆ ϩ ͙18

15. ͙8

16. ͙12

ෆ ϩ ͙27

ෆ Ϫ 3͙20

17. 2͙45

18. 3͙50

ෆ Ϫ 2͙32

ෆ Ϫ ͙8

19. ͙2

20. ͙20

ෆ Ϫ ͙125

46. ͙6

ෆ(͙15

ෆ Ϫ 1)

47. ͙3t

ෆ(͙9t

ෆ Ϫ ͙tෆ2 )

3

3

3

3

48. ͙2ෆ(͙12x

ෆ Ϫ ͙2x

ෆ)

3

3

21.

͙ෆ

45x 3 Ϫ ͙ෆ

18x 2 ϩ ͙ෆ

50x 2 Ϫ ͙ෆ

20x 3

49. (͙3ෆ ϩ 2)(͙3ෆ Ϫ 5)

22.

͙ෆ

12x Ϫ ͙18x

ෆ Ϫ ͙ෆ

300x ϩ ͙98x

50. (͙5ෆ ϩ 2)(͙5ෆ Ϫ 6)

5

5

23. 2͙24

ෆ ϩ ͙81

51. (͙11

ෆ Ϫ 3)(͙11

ෆ ϩ 3)

ෆ ϩ 2͙375

24. 5͙24

52. (͙2ෆ ϩ 5)(͙2ෆ ϩ 5)

25. ͙48

ෆ Ϫ 2͙243

53. (2͙5ෆ Ϫ 7)(2͙5

ෆ ϩ 4)

26. ͙64

ෆ ϩ 7͙2ෆ

54. (2͙6ෆ Ϫ 3)(2͙6

ෆ ϩ 4)

3

3

3

4

5

3

4

5

27.

3

3

͙

54t4y3 Ϫ ͙ෆ

16t4y3

55. (2͙3ෆ Ϫ ͙6

ෆ)(͙3ෆ ϩ 2͙6

ෆ)

28.

2 5

͙2000w

ෆ2ෆ

z5 Ϫ ͙16w

z

56. (3͙3ෆ Ϫ ͙2

ෆ)(͙2ෆ ϩ ͙3

ෆ)

3

3

3

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9.3

2

U3V Conjugates

94. (3͙a

ෆ ϩ 2)

Find the product of each pair of conjugates. See Example 5.

ϩ 2)

95. (1 ϩ ͙x

57. (͙3ෆ Ϫ 2)(͙3ෆ ϩ 2)

96. (͙x

Ϫ 1 ϩ 1)

2

2

ෆ )(7 ϩ ͙3

ෆ)

58. (7 Ϫ ͙3

97. ͙4w

ෆ Ϫ ͙9w

ෆ )(͙5ෆ Ϫ ͙2

ෆ)

59. (͙5ෆ ϩ ͙2

98. 10͙m

ෆ Ϫ ͙16m

ෆ )(͙6ෆ Ϫ ͙5

ෆ)

60. (͙6ෆ ϩ ͙5

99. 2͙aෆ3 ϩ 3͙aෆ3 Ϫ 2a͙4a

ෆ ϩ 1)(2͙5ෆ Ϫ 1)

61. (2͙5

2

2

2

100. 5͙w

y Ϫ 7͙w

y ϩ 6͙w

y

ෆ Ϫ 4)(3͙2ෆ ϩ 4)

62. (3͙2

͙xෆ5 ϩ 2x͙xෆ3

102. ͙8x

ෆ3 ϩ ͙50x

ෆ3 Ϫ x͙2x

3

3

4

103. ͙Ϫ16x

ෆ ϩ 5x͙54x

3

3

5 7

5 7

104. ͙3x

y Ϫ ͙24x

y

101.

ෆ ϩ ͙5

ෆ )(3͙2ෆ Ϫ ͙5

ෆ)

63. (3͙2

ෆ Ϫ ͙7

ෆ )(2͙3ෆ ϩ ͙7

ෆ)

64. (2͙3

65. (5 Ϫ 3͙xෆ )(5 ϩ 3͙xෆ )

66. (4͙yෆ ϩ 3͙zෆ )(4͙yෆ Ϫ 3͙zෆ )

105. ͙2x

ෆ и ͙2x

106. ͙2m

ෆ и ͙2n

3

U4V Multiplying Radicals with Different Indices

Write each product as a single radical expression.

See Example 6.

67. ͙3ෆ и ͙3

68. ͙3

ෆ и ͙3ෆ

69. ͙5ෆ и ͙5ෆ

70. ͙2ෆ и ͙2ෆ

71. ͙2ෆ и ͙5

72. ͙6ෆ и ͙2ෆ

73. ͙2ෆ и ͙3ෆ

74. ͙3ෆ и ͙2ෆ

3

3

4

4

3

3

3

5

3

4

585

3

4

3

4

Applications

Solve each problem.

107. Area of a rectangle. Find the exact area of a rectangle

that has a length of ͙6ෆ feet and a width of ͙3ෆ feet.

108. Volume of a cube. Find the exact volume of a cube with

sides of length ͙3ෆ meters.

Miscellaneous

Simplify each expression.

75. ͙300

ෆ ϩ ͙3

76. ͙50

ෆ ϩ ͙2

77. 2͙5

ෆ и 5͙6

78. 3͙6

ෆ и 5͙10

109. Area of a trapezoid. Find the exact area of a trapezoid with

ෆ feet and bases of ͙3

ෆ feet and ͙12

ෆ feet.

a height of ͙6

ͱ 3 ft

ෆ )(͙7ෆ Ϫ 2)

79. (3 ϩ 2͙7

80. (2 ϩ ͙7

ෆ )(͙7ෆ Ϫ 2)

ෆ и 4͙w

81. 4͙w

ෆ и 5͙m

82. 3͙m

83.

͙2t

ෆ и ͙10t

85. (2͙5

ෆ ϩ ͙2

ෆ)(3͙5ෆ Ϫ ͙2

ෆ)

86. (3͙2

ෆ Ϫ ͙3

ෆ)(2͙2ෆ ϩ 3͙3

ෆ)

84.

5

ͱ3 m

͙3x

ෆ3 и ͙6x

ෆ2

4

͙2ෆ ͙2ෆ

87. ᎏᎏ ϩ ᎏᎏ

3

5

͙2ෆ ͙3ෆ

88. ᎏᎏ ϩ ᎏᎏ

4

5

89. (5 ϩ 2͙2

ෆ)(5 Ϫ 2͙2

ෆ)

ෆ)(3 ϩ 2͙7

ෆ)

90. (3 Ϫ 2͙7

ͱ 6 ft

ͱ3 m

ͱ12 ft

ͱ3 m

Figure for Exercise 108

Figure for Exercise 109

110. Area of a triangle. Find the exact area of a triangle with

a base of ͙30

ෆ meters and a height of ͙6ෆ meters.

ͱ6 m

2

91. (3 ϩ ͙xෆ)

ͱ 30 m

2

92. (1 Ϫ ͙xෆ)

2

93. (5͙xෆ Ϫ 3)

Figure for Exercise 110

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Chapter 9 Radicals and Rational Exponents

Getting More Involved

113. Exploration

Because 3 is the square of ͙3ෆ, a binomial such as

y 2 Ϫ 3 is a difference of two squares.

111. Discussion

ෆ ϩ ͙b

ෆ ϭ ͙a

ϩ b for all values of a and b?

Is ͙a

a) Factor y 2 Ϫ 3 and 2a 2 Ϫ 7 using radicals.

b) Use factoring with radicals to solve the equation

x 2 Ϫ 8 ϭ 0.

c) Assuming a is a positive real number, solve the

equation x 2 Ϫ a ϭ 0.

112. Discussion

Which of the following equations are identities?

ෆ ϭ 3͙xෆ

a) ͙9x

b) ͙9

ϩ x ϭ 3 ϩ ͙xෆ

c) ͙xෆ

Ϫ 4 ϭ ͙xෆ Ϫ 2

d)

Ίᎏ๶4xᎏ ϭ ͙ᎏ2ᎏxෆ

Mid-Chapter Quiz

Sections 9.1 through 9.3

3

1. ͙64

2. ͙Ϫ27

3. ͙120

4. ͙56

12x7

5. ͙ෆ

6.

3

3 13

͙

24a

bෆ

8.

Ί๶

7.

Ί๶

3

17.

10. 1003͞2

11. Ϫ163͞2

51͞3

12. ᎏ

5Ϫ2͞3

΂

3

3

͙

8x5 ϩ ͙ෆ

27x5

Miscellaneous.

8x3

ᎏᎏ

9

9. 811͞2

ෆ)(8 Ϫ ͙10

ෆ)

16. (8 ϩ ͙10

15. 3͙10

ෆ и 2͙14

w

ᎏᎏ

16

Chapter 9

18. Find the domain of the expression ͙6

Ϫ 3x

ෆ.

19. Find the solution set to ͙ෆ

x2 ϭ x.

΃

Ϫ3

20. Find the solution set to (x4)1͞4 ϭ ͉ x͉.

3

ෆ и ͙2ෆ as a single radical expression.

21. Write the product ͙2

Perform the indicated operations.

13. 2͙3

ෆ Ϫ 5͙6

ෆ Ϫ 4͙3

ෆ ϩ ͙6

ෆ 14. 9͙20

ෆ Ϫ 3͙45

22. Suppose that h(t) ϭ 5t2͞3. Find h(8).

9.4

In This Section

U1V Rationalizing the

Denominator

2

U4V Rationalizing Denominators

Using Conjugates

Expressions

Quotients, Powers, and Rationalizing

Denominators

In this section, we will continue studying operations with radicals. We will first learn

how to rationalize denominators, and then we will find quotients and powers with

U1V Rationalizing the Denominator

Square roots such as ͙2ෆ, ͙3

ෆ, and ͙5

ෆ are irrational numbers. If roots of this type

appear in the denominator of a fraction, it is customary to rewrite the fraction with a

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9-31

9.4

Quotients, Powers, and Rationalizing Denominators

587

rational number in the denominator, or rationalize it. We rationalize a denominator by

multiplying both the numerator and denominator by another radical that makes the

denominator rational.

You can find products of radicals in two ways. By definition, ͙2ෆ is the positive

number that you multiply by itself to get 2. So,

ෆ и ͙2

ෆ ϭ 2.

͙2

3

3

3

ෆ ϭ ͙4

ෆ ϭ 2. Note that ͙

2ෆ и ͙

2ෆ ϭ ͙

4ෆ by the product

By the product rule, ͙2ෆ и ͙2

3

rule, but ͙4ෆ 2. By definition of a cube root,

3

3

3

2ෆ и ͙

2ෆ и ͙

2ෆ ϭ 2.

͙

E X A M P L E

1

Rationalizing the denominator

Rewrite each expression with a rational denominator.

͙3ෆ

a) ᎏ

͙5ෆ

If you are going to compute the

value of a radical expression with a

calculator, it does not matter if the

denominator is rational. However,

rationalizing the denominator provides another opportunity to practice building up the denominator of

3

b) ᎏ

3

͙

2ෆ

Solution

a) Because ͙5ෆ и ͙5

ෆ ϭ 5, multiplying both the numerator and denominator by ͙5

will rationalize the denominator:

15

͙3ෆ ϭ ͙3ෆ и ͙5ෆ ϭ ͙

ᎏᎏ By the product rule, ͙3ෆ и ͙5ෆ ϭ ͙15

ෆ.

ᎏ ᎏ

5

͙5ෆ

͙5ෆ ͙5ෆ

b) We must build up the denominator to be the cube root of a perfect cube. So we

3

3

3

3

multiply by ͙

ෆ4 to get ͙

ෆ4 и ͙

ෆ2 ϭ ͙

ෆ8:

3

3

3

3 ͙

3

4ෆ

3͙4ෆ

3͙4ෆ

ϭ

и

ϭ

ϭ

3

3

3

3

͙2ෆ ͙

͙

2ෆ

4ෆ

͙8ෆ

2

Now do Exercises 1–8

CAUTION To rationalize a denominator with a single square root, you simply

multiply by that square root. If the denominator has a cube root, you build

the denominator to a cube root of a perfect cube, as in Example 1(b). For a

fourth root you build to a fourth root of a perfect fourth power, and so on.

When simplifying a radical expression, we have three specific conditions to satisfy.

First, we use the product rule to factor out perfect nth powers from the radicand in nth

roots. That is, we factor out perfect squares in square roots, perfect cubes in cube roots,

and so on. For example,

ෆ ϭ 6͙2

͙72

ෆ ϭ ͙36

ෆ и ͙2

and

3

3

3

3

͙

ෆϭ͙

24

8ෆ и ͙

3ෆ ϭ 2͙

3ෆ.