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3 Adding, Subtracting, and Multiplying Radicals

3 Adding, Subtracting, and Multiplying Radicals

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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

ෆ Simplify each radical.

ϭ 5͙2





Add like radicals.



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

ෆ Simplify each radical.

ϭ 16x͙2x

ෆ Ϫ 2x



c)



Add like radicals only.



͙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



Simplify each radical.



ϭ Ϫxy͙2x



3



Now do Exercises 13–28



U2V Multiplying Radicals

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.



U Helpful Hint V

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





Product rule for radicals



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



Product rule for radicals



ϭ ͙ෆ

9a и ͙2a



2



ϭ 3a͙2a





Simplify.



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9-25



9.3



c)



Adding, Subtracting, and Multiplying Radicals



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



Product rule for radicals



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



Multiplying radicals

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

ෆ Combine like radicals.

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

radicals with different indices we convert the radicals into exponential expressions

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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9.3



583



Adding, Subtracting, and Multiplying Radicals



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



Multiplying radicals with different indices

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



Write in radical notation.



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

33 Write in radical notation.

6



6



ϭ ͙ෆ

22 и 33



Product rule for radicals



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

cannot add the exponents [Example 6(b)]. Instead, we write each factor

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

radicand.

2. The

property is used to combine like

radicals.

3. The product rule for radicals is used to multiply radicals

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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9/22/10



10:45 AM



Page 584



Exercises



584



9-28



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.



U1V Adding and Subtracting Radicals



U2V Multiplying Radicals



All variables in the following exercises represent positive

numbers. Simplify the sums and differences. Give exact

answers. See Example 1.



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



Adding, Subtracting, and Multiplying Radicals

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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586



9-30



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?

Explain your answers.

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





Simplify each radical expression.



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

U V Simplifying Radicals

U3V Dividing Radicals

U4V Rationalizing Denominators

Using Conjugates

U5V Powers of Radical

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

radicals.



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



U Helpful Hint V

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

a fraction and multiplying radicals.



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.



U2V Simplifying Radicals

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.



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