Lesson 10-2 Simplifying Radicals 619

Simplifying Radicals10-2

Objective To simplify radicals involving products and quotients

Suppose you are bringing a mirror into your living room. What is the maximum height of a square mirror that will fit through the doorway shown? Justify your reasoning.

In the Solve It, the maximum height of the mirror is a radical expression. A radical expression, such as 2 !3 or !x 1 3, is an expression that contains a radical. A radical expression is simplified if the following statements are true.

• The radicand has no perfect-square factors other than 1. • The radicand contains no fractions. • No radicals appear in the denominator of a fraction.

Simplified Not Simplified

3 !5 9 !x !24 3 !12 Å

x2 5

!7

Essential Understanding You can simplify radical expressions using multiplication and division properties of square roots.

Property Multiplication Property of Square Roots

Algebra Example

For a $ 0 and b $ 0, !ab 5 !a ? !b. !48 5 !16 ? !3 5 4 !3

You can use the Multiplication Property of Square Roots to simplify radicals by removing perfect-square factors from the radicand.

Lesson Vocabulary

•radical expression•rationalize the

denominator

LessonVocabulary

w

2w

Dynamic ActivitySimplifying Radicals

AC T I V I T I

E S

DYNAMIC Dynamic Activity

Use what you know about triangles to solve this problem.

MATHEMATICAL PRACTICES

Content StandardPrepares for A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

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Problem 1

Problem 3

Problem 2

Got It?

Got It?

620 Chapter 10 Radical Expressions and Equations

Removing Perfect-Square Factors

What is the simplified form of !160?

!160 5 !16 ? 10 16 is the greatest perfect-square factor of 160.

5 !16 ? !10 Use the Multiplication Property of Square Roots.

5 4 !10 Simplify !16.

1. What is the simplified form of !72?

Sometimes you can simplify radical expressions that contain variables. A variable with an even exponent is a perfect square. A variable with an odd exponent is the product of

a perfect square and the variable. For example, n3 5 n2 ? n, so "n3 5 "n2 ? n. In this lesson, assume that all variables in radicands represent nonnegative numbers.

Removing Variable Factors

Multiple Choice What is the simplified form of "54n7?

n3 !54n 9n6

!6n 3n3 !6n 3n !27n

"54n7 5 "9n6 ? 6n 9n6, or (3n3)2, is a perfect-square factor of 54n7.

5 "9n6 ? !6n Use the Multiplication Property of Square Roots.

5 3n3 !6n Simplify "9n6.

The correct answer is C.

2. What is the simplified form of 2m"80m9?

You can use the Multiplication Property of Square Roots to write !a ? !b 5 !ab.

Multiplying Two Radical Expressions

What is the simplified form of 2!7t ? 3"14t

2?

2!7t ? 3 "14t2 5 6 "7t ? 14t2 Multiply the whole numbers and use the Multiplication Property of Square Roots.

5 6"98t3 Simplify under the radical symbol.

5 6"49t2 ? 2t 49t2, or (7t)2, is a perfect-square factor of 98t3.

5 6"49t2 ? !2t Use the Multiplication Property of Square Roots.

5 6 ? 7t !2t Simplify "49t2 .

5 42t !2t Simplify.

What strategy can you use to find the factor to remove?You can solve a simpler problem by first just listing the factors of the radicand. Then choose the greatest perfect square on the list.

How is this problem similar to Problem 1?In both problems, you need to remove a perfect-square factor from the radicand. In this problem, however, the factor you remove contains a variable.

What property allows you to multiply the whole numbers first?The Commutative Property of Multiplication allows you to change the order of the factors.

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Problem 4

Got It?

Got It?

Lesson 10-2 Simplifying Radicals 621

How is this like problems you have done before?The width and height of the door are two legs of a right triangle. This is like finding the hypotenuse of a right triangle using the Pythagorean Theorem.

3. What is the simplified form of each expression in parts (a)–(c)?

a. 3 !6 ? !18 b. !2a ? "9a3 c. 7 !5x ? 3 "20x

5

d. Reasoning In Problem 3, can you simplify the given product by first simplifying "14t2? Explain.

Writing a Radical Expression

Art A rectangular door in a museum is three times as tall as it is wide. What is a simplified expression for the maximum length of a painting that fits through the door?

Use the Pythagorean Theorem.

The diagonal length d of the doorway

The door is w units wide and 3w units high.

d2 5 w2 1 (3w)2 Pythagorean Theorem

d2 5 w2 1 9w2 Simplify (3w)2.

d2 5 10w2 Combine like terms.

d 5 "10w2 Find the principal square root of each side.

d 5 "w2 ? !10 Multiplication Property of Square Roots

d 5 w !10 Simplify "w2.

An expression for the maximum length of the painting is w !10, or about 3.16w.

4. A door’s height is four times its width w. What is the maximum length of a painting that fits through the door?

You can simplify some radical expressions using the following property.

3w

w

Property Division Property of Square Roots

Algebra Example

For a $ 0 and b . 0, Åab 5

!a!b

. Å3649 5

!36!49

567

When a radicand has a denominator that is a perfect square, it is easier to apply the Division Property of Square Roots first and then simplify the numerator and denominator of the result. When the denominator of a radicand is not a perfect square, it may be easier to simplify the fraction first.

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Problem 6

Got It?

Problem 5

Got It?

622 Chapter 10 Radical Expressions and Equations

Simplifying Fractions Within Radicals

What is the simplified form of each radical expression?

A Å6449

Å6449 5

!64!49

Use the Division Property of Square Roots.

5 87 Simplify !64 and !49.

B Å8x3

50x

Å8x3

50x 5 Å4x2

25 Divide the numerator and denominator by 2x.

5 "4x2

!25 Use the Division Property of Square Roots.

5 !4 ? "x2

!25 Use the Multiplication Property of Square Roots.

5 2x5 Simplify !4, "x2, and !25.

5. What is the simplified form of each radical expression?

a. Å144

9 b. Å36a4a3 c. Å

25y3

z2

When a radicand in a denominator is not a perfect square, you may need to rationalize the denominator to remove the radical. To do this, multiply the numerator and denominator by the same radical expression. Choose an expression that makes the radicand in the denominator a perfect square. It may be helpful to start by simplifying the original radical in the denominator.

Rationalizing Denominators

What is the simplified form of each expression?

A !3!7

B !7!8n

!3!7

5!3!7

?!7!7

!7!8n

5!7

2!2n

5!21!49

5!7

2!2n?!2n!2n

5!21

7 5!14n

2"4n2

5!14n

4n

6. What is the simplified form of each radical expression?

a. !2!3

b. !5!18m

c. Å7s3

Which method should you use?If the denominator is a perfect square, apply the Division Property of Square Roots first. If not, simplify the fraction first.

Does multiplying an expression by Á 7

Á 7

change its value?

No. The fraction Á7Á7 is equal to 1. Multiplying an expression by 1 won’t change its value.

Multiply by !2n!2n

.

Multiply by !7!7

.

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Lesson Check

Lesson 10-2 Simplifying Radicals 623

Do you know HOW?Simplify each radical expression.

1. !98 2. Í16b5

3. 3!5m ? 4Å15m3 4. Å

15xx3

5. !5!3

6. !6!2n

Do you UNDERSTAND? 7. Vocabulary Is the radical expression in simplified

form? Explain.

a. !313 b. 7Å

611 c. 25 !175

8. Compare and Contrast Simplify 3!12

two different ways. Which way do you prefer? Explain.

9. Writing Explain how you can tell whether a radical expression is in simplified form.

Practice and Problem-Solving Exercises

Simplify each radical expression.

10. !225 11. !99 12. !128 13. 2!60

14. 24 !117 15. 5 !700 16. "192s2 17. "50t

5

18. 3 "18a2 19. 221 "27x9 20. 3 "150b8 21. 22 "243y

3

Simplify each product.

22. !8 ? !32 23. 13 !6 ? !24 24. 4 !10 ? 2 !90

25. 5 !6 ? 16 !216 26. 25 !21 ? (23 !42) 27. !18n ? "98n3

28. 3 !5c ? 7 "15c2 29. !2y ? "128y5 30. 26 "15s3 ? 2 !75

31. 29 "28a2 ?13 !63a 32. 10 "12x

3 ? 2 "6x

3 33. 213 "18c5 ? Q26 "8c9R

34. Construction Students are building rectangular wooden frames for the set of a school play. The height of a frame is 6 times the width w. Each frame has a brace that connects two opposite corners of the frame. What is a simplified expression for the length of a brace?

35. Park A park is shaped like a rectangle with a length 5 times its width w. What is a simplified expression for the distance between opposite corners of the park?

Simplify each radical expression.

36. Å1625 37. 7 Å

632 38. 24 Å

100729 39. Å

3x3

64x2

40. 25 Å162t3

2t 41. 11Å

49a5

4a3 42. 1!11

43. !5!8x

44. 3 !6!15

45. 22!11

46. 2!24

"48t4 47. 8!7s

"28s3

PracticeA See Problems 1 and 2.

See Problem 3.

See Problem 4.

See Problems 5 and 6.

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES

STEM

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624 Chapter 10 Radical Expressions and Equations

48. Look for a Pattern From a viewing height of h feet, the approximate distance d to the horizon, in miles, is

given by the equation d 5 Å3h2 .

a. To the nearest mile, what is the distance to the horizon from a height of 150 ft? 225 ft? 300 ft?

b. How does the distance to the horizon increase as the height increases?

49. Think About a Plan A square picture on the front page of a newspaper occupies an area of 24 in.2. What is the length of each side of the picture? Write your answer as a radical in simplified form.

• How can you find the side length of a square if you know the area? • What property can you use to write your answer in simplified form?

Explain why each radical expression is or is not in simplified form.

50. 13x!4

51. 3!3

52. 24!5 53. 5!30

54. Error Analysis A student simplified the radical expression at the right. What mistake did the student make? What is the correct answer?

55. Reasoning You can simplify radical expressions with negative exponents by first rewriting the expressions using positive exponents. What are the simplified forms of the following radical expressions?

a. !3

"f 23 b. "x23

!x c. "5a22

"10a21 d.

"(2m)23

m21

56. Sports The bases in a softball diamond are located at the corners of a

3600-ft2 square. How far is a throw from second base to home plate?

57. Suppose a and b are positive integers.

a. Verify that if a 5 18 and b 5 10, then !a ? !b 5 6!5.

b. Open-Ended Find two other pairs of positive integers a and b such that

!a ? !b 5 6!5.

Simplify each radical expression.

58. !12 ? !75 59. !26 ? 2 60. !72!64

61. 22

"a3

62. !180!3

63. "x2

"y3 64. 23!2

!6 65. !8 ? !10

66. "20a2b3 67. "a3b5c3 68. Å3m

16m2 69. 16a

"6a3

Solve each equation. Leave your answer in simplified radical form.

70. x2 1 6x 2 9 5 0 71. n2 2 2n 1 1 5 5 72. 3y2 2 4y 2 2 5 0

73. Open-Ended What are three numbers whose square roots can be written in the

form a!3 for some integer value of a?

ApplyB

√ x5x25

= √55

Home plate

Firstbase

Secondbase

Thirdbase

hsm11a1se_1002_t01998

h

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Lesson 10-2 Simplifying Radicals 625

Mixed Review

Determine whether the given lengths can be side lengths of a right triangle.

83. 7, 24, 25 84. 1, 43, 53 85. 5, 13, 14

Factor each expression.

86. 64y2 2 9 87. a2 2 81 88. 25 2 16b2

Get Ready! To prepare for Lesson 10-3, do Exercises 89–91.

Simplify each product.

89. (3a 2 4)(2a 1 1) 90. (2m 2 3n)(4n 2 2m) 91. (5 1 2x)(2x 1 3)

See Lesson 10-1.

See Lesson 8-7.

See Lesson 8-3.

Simplify each radical expression.

74. !24 ? !2x ? !3x 75. 2b(!5b)2 76. "45a7 ? !20a

77. Geometry The equation r 5 ÅAp gives the radius r of a circle with area A. What is

the radius of a circle with the given area? Write your answer as a simplified radical and as a decimal rounded to the nearest hundredth.

a. 50 ft2 b. 32 in.2 c. 10 m2

78. For a linear equation in standard form Ax 1 By 5 C, where A 2 0 and B 2 0, the

distance d between the x- and y-intercepts is given by d 5 ÅQCAR

21 QC

BR2

. What is

the distance between the x- and y-intercepts of the graph of 4x 2 3y 5 2?

ChallengeC

Standardized Test Prep

79. What is the simplified form of "12y5?

2"3y5 4y4!3y 2y2!3y 3y3

80. In the proportion 3b 5

78 2 b , what is the value of b?

6 218 12

5 512

81. The area of the triangle at the right is 24 in.2. What is the height of the triangle?

1.8 in. 7 in.

3 in. 16 in.

82. An architect is sketching a line on a coordinate grid showing the location of a pipe. The line has an x-intercept of 22 and a y-intercept of 3. What is an equation of the architect’s line?

SAT/ACT

hsm11a1se_1002_t13237.ai

2x 2

x 4

ShortResponse

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