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Elementary Algebra

9.2 Simplify Square Roots

Elementary Algebra9.2 Simplify Square Roots
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Learning Objectives

By the end of this section, you will be able to:

  • Use the Product Property to simplify square roots
  • Use the Quotient Property to simplify square roots

Be Prepared 9.2

Before you get started take this readiness quiz.

  1. Simplify: 8017680176.
    If you missed this problem, review Example 1.65.
  2. Simplify: n9n3n9n3.
    If you missed this problem, review Example 6.59.
  3. Simplify: q4q12q4q12.
    If you missed this problem, review Example 6.60.

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that 5050 is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use Figure 9.2.

But what if we want to estimate 500500? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.

A square root is considered simplified if its radicand contains no perfect square factors.

Simplified Square Root

aa is considered simplified if aa has no perfect square factors.

So 3131 is simplified. But 3232 is not simplified, because 16 is a perfect square factor of 32.

Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that (ab)m=ambm(ab)m=ambm. The corresponding property of square roots says that ab=a·bab=a·b.

Product Property of Square Roots

If a, b are non-negative real numbers, then ab=a·bab=a·b.

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in Example 9.12.

Example 9.12

How To Use the Product Property to Simplify a Square Root

Simplify: 5050.

Try It 9.23

Simplify: 4848.

Try It 9.24

Simplify: 4545.

Notice in the previous example that the simplified form of 5050 is 5252, which is the product of an integer and a square root. We always write the integer in front of the square root.

How To

Simplify a square root using the product property.

  1. Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
  2. Step 2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Step 3. Simplify the square root of the perfect square.

Example 9.13

Simplify: 500500.

Try It 9.25

Simplify: 288288.

Try It 9.26

Simplify: 432432.

We could use the simplified form 105105 to estimate 500500. We know 5 is between 2 and 3, and 500500 is 105105. So 500500 is between 20 and 30.

The next example is much like the previous examples, but with variables.

Example 9.14

Simplify: x3x3.

Try It 9.27

Simplify: b5b5.

Try It 9.28

Simplify: p9p9.

We follow the same procedure when there is a coefficient in the radical, too.

Example 9.15

Simplify: 25y5.25y5.

Try It 9.29

Simplify: 16x716x7.

Try It 9.30

Simplify: 49v949v9.

In the next example both the constant and the variable have perfect square factors.

Example 9.16

Simplify: 72n772n7.

Try It 9.31

Simplify: 32y532y5.

Try It 9.32

Simplify: 75a975a9.

Example 9.17

Simplify: 63u3v563u3v5.

Try It 9.33

Simplify: 98a7b598a7b5.

Try It 9.34

Simplify: 180m9n11180m9n11.

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify 25+14425+144 we must simplify each square root separately first, then add to get the sum of 17.

The expression 17+717+7 cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.

In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.

Example 9.18

Simplify: 3+323+32.

Try It 9.35

Simplify: 5+755+75.

Try It 9.36

Simplify: 2+982+98.

The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

Example 9.19

Simplify: 44824482.

Try It 9.37

Simplify: 1075510755.

Try It 9.38

Simplify: 64536453.

Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

Example 9.20

Simplify: 964964.

Try It 9.39

Simplify: 25162516.

Try It 9.40

Simplify: 49814981.

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

Example 9.21

Simplify: 45804580.

Try It 9.41

Simplify: 75487548.

Try It 9.42

Simplify: 9816298162.

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, aman=amn,a0aman=amn,a0.

Example 9.22

Simplify: m6m4m6m4.

Try It 9.43

Simplify: a8a6a8a6.

Try It 9.44

Simplify: x14x10x14x10.

Example 9.23

Simplify: 48p73p348p73p3.

Try It 9.45

Simplify: 75x53x75x53x.

Try It 9.46

Simplify: 72z122z1072z122z10.

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

(ab)m=ambm,b0(ab)m=ambm,b0

We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

Quotient Property of Square Roots

If a, b are non-negative real numbers and b0b0, then

ab=abab=ab

Example 9.24

Simplify: 21642164.

Try It 9.47

Simplify: 19491949.

Try It 9.48

Simplify: 28812881.

Example 9.25

How to Use the Quotient Property to Simplify a Square Root

Simplify: 27m319627m3196.

Try It 9.49

Simplify: 24p34924p349.

Try It 9.50

Simplify: 48x510048x5100.

How To

Simplify a square root using the quotient property.

  1. Step 1. Simplify the fraction in the radicand, if possible.
  2. Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
  3. Step 3. Simplify the radicals in the numerator and the denominator.

Example 9.26

Simplify: 45x5y445x5y4.

Try It 9.51

Simplify: 80m3n680m3n6.

Try It 9.52

Simplify: 54u7v854u7v8.

Be sure to simplify the fraction in the radicand first, if possible.

Example 9.27

Simplify: 81d925d481d925d4.

Try It 9.53

Simplify: 64x79x364x79x3.

Try It 9.54

Simplify: 16a9100a516a9100a5.

Example 9.28

Simplify: 18p5q732pq218p5q732pq2.

Try It 9.55

Simplify: 50x5y372x4y50x5y372x4y.

Try It 9.56

Simplify: 48m7n2125m5n948m7n2125m5n9.

Section 9.2 Exercises

Practice Makes Perfect

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

53.

27 27

54.

80 80

55.

125 125

56.

96 96

57.

200 200

58.

147 147

59.

450 450

60.

252 252

61.

800 800

62.

288 288

63.

675 675

64.

1250 1250

65.

x 7 x 7

66.

y 11 y 11

67.

p 3 p 3

68.

q 5 q 5

69.

m 13 m 13

70.

n 21 n 21

71.

r 25 r 25

72.

s 33 s 33

73.

49 n 17 49 n 17

74.

25 m 9 25 m 9

75.

81 r 15 81 r 15

76.

100 s 19 100 s 19

77.

98 m 5 98 m 5

78.

32 n 11 32 n 11

79.

125 r 13 125 r 13

80.

80 s 15 80 s 15

81.

200 p 13 200 p 13

82.

128 q 3 128 q 3

83.

242 m 23 242 m 23

84.

175 n 13 175 n 13

85.

147 m 7 n 11 147 m 7 n 11

86.

48 m 7 n 5 48 m 7 n 5

87.

75 r 13 s 9 75 r 13 s 9

88.

96 r 3 s 3 96 r 3 s 3

89.

300 p 9 q 11 300 p 9 q 11

90.

192 q 3 r 7 192 q 3 r 7

91.

242 m 13 n 21 242 m 13 n 21

92.

150 m 9 n 3 150 m 9 n 3

93.

5 + 12 5 + 12

94.

8 + 96 8 + 96

95.

1 + 45 1 + 45

96.

3 + 125 3 + 125

97.

10 24 2 10 24 2

98.

8 80 4 8 80 4

99.

3 + 90 3 3 + 90 3

100.

15 + 75 5 15 + 75 5

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

101.

49 64 49 64

102.

100 36 100 36

103.

121 16 121 16

104.

144 169 144 169

105.

72 98 72 98

106.

75 12 75 12

107.

45 125 45 125

108.

300 243 300 243

109.

x 10 x 6 x 10 x 6

110.

p 20 p 10 p 20 p 10

111.

y 4 y 8 y 4 y 8

112.

q 8 q 14 q 8 q 14

113.

200 x 7 2 x 3 200 x 7 2 x 3

114.

98 y 11 2 y 5 98 y 11 2 y 5

115.

96 p 9 6 p 96 p 9 6 p

116.

108 q 10 3 q 2 108 q 10 3 q 2

117.

36 35 36 35

118.

144 65 144 65

119.

20 81 20 81

120.

21 196 21 196

121.

96 x 7 121 96 x 7 121

122.

108 y 4 49 108 y 4 49

123.

300 m 5 64 300 m 5 64

124.

125 n 7 169 125 n 7 169

125.

98 r 5 100 98 r 5 100

126.

180 s 10 144 180 s 10 144

127.

28 q 6 225 28 q 6 225

128.

150 r 3 256 150 r 3 256

129.

75 r 9 s 8 75 r 9 s 8

130.

72 x 5 y 6 72 x 5 y 6

131.

28 p 7 q 2 28 p 7 q 2

132.

45 r 3 s 10 45 r 3 s 10

133.

100 x 5 36 x 3 100 x 5 36 x 3

134.

49 r 12 16 r 6 49 r 12 16 r 6

135.

121 p 5 81 p 2 121 p 5 81 p 2

136.

25 r 8 64 r 25 r 8 64 r

137.

32 x 5 y 3 18 x 3 y 32 x 5 y 3 18 x 3 y

138.

75 r 6 s 8 48 r s 4 75 r 6 s 8 48 r s 4

139.

27 p 2 q 108 p 5 q 3 27 p 2 q 108 p 5 q 3

140.

50 r 5 s 2 128 r 2 s 5 50 r 5 s 2 128 r 2 s 5

Everyday Math

141.
  1. Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify 288288 to determine the length and the width of his garden. Round to the nearest tenth of a foot.
  2. Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify 28852885 to determine the length and width of the new garden. Round to the nearest tenth of a foot.
142.
  1. Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify 64166416 to determine the number of seconds it takes for the sunglasses to reach the ground.
  2. Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify 1441614416 to determine the number of seconds it takes for the sunglasses to reach the ground.

Writing Exercises

143.

Explain why x4=x2x4=x2. Then explain why x16=x8x16=x8.

144.

Explain why 7+97+9 is not equal to 7+97+9.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and three rows. The columns are labeled, “I can…,” “confidently,” “with some help,” and “no—I don’t get it!” The rows under “I can…” Read, “use the Product Property to simplify square roots.,” and “use the Quotient Property to simplify square roots.” The other rows unders the other columns are blank.

After reviewing this checklist, what will you do to become confident for all objectives?

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