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Intermediate Algebra 2e

8.2 Simplify Radical Expressions

Intermediate Algebra 2e8.2 Simplify Radical Expressions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

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

  • Use the Product Property to simplify radical expressions
  • Use the Quotient Property to simplify radical expressions
Be Prepared 8.4

Before you get started, take this readiness quiz.

Simplify: x9x4.x9x4.
If you missed this problem, review Example 5.13.

Be Prepared 8.5

Simplify: y3y11.y3y11.
If you missed this problem, review Example 5.13.

Be Prepared 8.6

Simplify: (n2)6.(n2)6.
If you missed this problem, review Example 5.17.

Use the Product Property to Simplify Radical Expressions

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.

A radical expression, an,an, is considered simplified if it has no factors of mn.mn. So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.

Simplified Radical Expression

For real numbers a and m, and n2,n2,

anis considered simplified ifahas no factors ofmnanis considered simplified ifahas no factors ofmn

For example, 55 is considered simplified because there are no perfect square factors in 5. But 1212 is not simplified because 12 has a perfect square factor of 4.

Similarly, 4343 is simplified because there are no perfect cube factors in 4. But 243243 is not simplified because 24 has a perfect cube factor of 8.

To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that (ab)n=anbn.(ab)n=anbn. The corresponding of Product Property of Roots says that abn=an·bn.abn=an·bn.

Product Property of nth Roots

If anan and bnbn are real numbers, and n2n2 is an integer, then

abn=an·bnandan·bn=abnabn=an·bnandan·bn=abn

We use the Product Property of Roots to remove all perfect square factors from a square root.

Example 8.13

Simplify Square Roots Using the Product Property of Roots

Simplify: 98.98.

Try It 8.25

Simplify: 48.48.

Try It 8.26

Simplify: 45.45.

Notice in the previous example that the simplified form of 9898 is 72,72, which is the product of an integer and a square root. We always write the integer in front of the square root.

Be careful to write your integer so that it is not confused with the index. The expression 7272 is very different from 27.27.

How To

Simplify a radical expression using the Product Property.

  1. Step 1. Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
  2. Step 2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Step 3. Simplify the root of the perfect power.

We will apply this method in the next example. It may be helpful to have a table of perfect squares, cubes, and fourth powers.

Example 8.14

Simplify: 500500 163163 2434.2434.

Try It 8.27

Simplify: 288288 813813 644.644.

Try It 8.28

Simplify: 432432 62536253 7294.7294.

The next example is much like the previous examples, but with variables. Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Example 8.15

Simplify: x3x3 x43x43 x74.x74.

Try It 8.29

Simplify: b5b5 y64y64 z53z53

Try It 8.30

Simplify: p9p9 y85y85 q136q136

We follow the same procedure when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.

Example 8.16

Simplify: 72n772n7 24x7324x73 80y144.80y144.

Try It 8.31

Simplify: 32y532y5 54p10354p103 64q104.64q104.

Try It 8.32

Simplify: 75a975a9 128m113128m113 162n74.162n74.

In the next example, we continue to use the same methods even though there are more than one variable under the radical.

Example 8.17

Simplify: 63u3v563u3v5 40x4y5340x4y53 48x4y74.48x4y74.

Try It 8.33

Simplify: 98a7b598a7b5 56x5y4356x5y43 32x5y84.32x5y84.

Try It 8.34

Simplify: 180m9n11180m9n11 72x6y5372x6y53 80x7y44.80x7y44.

Example 8.18

Simplify: −273−273 −164.−164.

Try It 8.35

Simplify: −643−643 −814.−814.

Try It 8.36

Simplify: −6253−6253 −3244.−3244.

We have seen how to use the order of operations to simplify some expressions with radicals. 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 since one term contains a radical and the other does not. The next example also 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 8.19

Simplify: 3+323+32 4482.4482.

Try It 8.37

Simplify: 5+755+75 1075510755

Try It 8.38

Simplify: 2+982+98 64536453

Use the Quotient Property to Simplify Radical Expressions

Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.

Example 8.20

Simplify: 45804580 1654316543 5804.5804.

Try It 8.39

Simplify: 75487548 542503542503 321624.321624.

Try It 8.40

Simplify: 9816298162 243753243753 43244.43244.

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 8.21

Simplify: m6m4m6m4 a8a53a8a53 a10a24.a10a24.

Try It 8.41

Simplify: a8a6a8a6 x7x34x7x34 y17y54.y17y54.

Try It 8.42

Simplify: x14x10x14x10 m13m73m13m73 n12n25.n12n25.

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 root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect power of the index, we simplify the numerator and denominator separately.

Quotient Property of Radical Expressions

If anan and bnbn are real numbers,b0,b0, and for any integer n2n2 then,

abn=anbnandanbn=abnabn=anbnandanbn=abn

Example 8.22

How to Simplify the Quotient of Radical Expressions

Simplify: 27m3196.27m3196.

Try It 8.43

Simplify: 24p349.24p349.

Try It 8.44

Simplify: 48x5100.48x5100.

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 8.23

Simplify: 45x5y445x5y4 24x7y3324x7y33 48x10y84.48x10y84.

Try It 8.45

Simplify: 80m3n680m3n6 108c10d63108c10d63 80x10y44.80x10y44.

Try It 8.46

Simplify: 54u7v854u7v8 40r3s6340r3s63 162m14n124.162m14n124.

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

Example 8.24

Simplify: 18p5q732pq218p5q732pq2 16x5y754x2y2316x5y754x2y23 5a8b680a3b24.5a8b680a3b24.

Try It 8.47

Simplify: 50x5y372x4y50x5y372x4y 16x5y754x2y2316x5y754x2y23 5a8b680a3b24.5a8b680a3b24.

Try It 8.48

Simplify: 48m7n2100m5n848m7n2100m5n8 54x7y5250x2y2354x7y5250x2y23 32a9b7162a3b34.32a9b7162a3b34.

In the next example, there is nothing to simplify in the denominators. Since the index on the radicals is the same, we can use the Quotient Property again, to combine them into one radical. We will then look to see if we can simplify the expression.

Example 8.25

Simplify: 48a73a48a73a −108323−108323 96x743x24.96x743x24.

Try It 8.49

Simplify: 98z52z98z52z −500323−500323 486m1143m54.486m1143m54.

Try It 8.50

Simplify: 128m92m128m92m −192333−192333 324n742n34.324n742n34.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with simplifying radical expressions.

Section 8.2 Exercises

Practice Makes Perfect

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

55.

2727

56.

8080

57.

125125

58.

9696

59.

147147

60.

450450

61.

800800

62.

675675

63.

324324 645645

64.

62536253 12861286

65.

644644 25632563

66.

3125431254 813813

In the following exercises, simplify using absolute value signs as needed.

67.

y11y11 r53r53 s104s104

68.

m13m13 u75u75 v116v116

69.

n21n21 q83q83 n108n108

70.

r25r25 p85p85 m54m54

71.

125r13125r13 108x53108x53 48y6448y64

72.

80s1580s15 96a7596a75 128b76128b76

73.

242m23242m23 405m104405m104 160n85160n85

74.

175n13175n13 512p55512p55 324q74324q74

75.

147m7n11147m7n11 48x6y7348x6y73 32x5y4432x5y44

76.

96r3s396r3s3 80x7y6380x7y63 80x8y9480x8y94

77.

192q3r7192q3r7 54m9n10354m9n103 81a9b8481a9b84

78.

150m9n3150m9n3 81p7q8381p7q83 162c11d124162c11d124

79.

−8643−8643 −2564−2564

80.

−4865−4865 −646−646

81.

−325−325 −18−18

82.

−83−83 −164−164

83.

5+125+12 1024210242

84.

8+968+96 88048804

85.

1+451+45 3+9033+903

86.

3+1253+125 15+75515+755

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

87.

45804580 82738273 18141814

88.

72987298 2481324813 69646964

89.

1003610036 813753813753 1256412564

90.

1211612116 162503162503 321624321624

91.

x10x6x10x6 p11p23p11p23 q17q134q17q134

92.

p20p10p20p10 d12d75d12d75 m12m48m12m48

93.

y4y8y4y8 u21u115u21u115 v30v126v30v126

94.

q8q14q8q14 r14r53r14r53 c21c94c21c94

95.

96x712196x7121

96.

108y449108y449

97.

300m564300m564

98.

125n7169125n7169

99.

98r510098r5100

100.

180s10144180s10144

101.

28q622528q6225

102.

150r3256150r3256

103.

75r9s875r9s8 54a8b3354a8b33 64c5d4464c5d44

104.

72x5y672x5y6 96r11s5596r11s55 128u7v126128u7v126

105.

28p7q228p7q2 81s8t3381s8t33 64p15q12464p15q124

106.

45r3s1045r3s10 625u10v33625u10v33 729c21d84729c21d84

107.

32x5y318x3y32x5y318x3y 5x6y940x5y335x6y940x5y33 5a8b680a3b245a8b680a3b24

108.

75r6s848rs475r6s848rs4 24x8y481x2y324x8y481x2y3 32m9n2162mn2432m9n2162mn24

109.

27p2q108p4q327p2q108p4q3 16c5d7250c2d2316c5d7250c2d23 2m9n7128m3n62m9n7128m3n6

110.

50r5s2128r2s650r5s2128r2s6 24m9n7375m4n324m9n7375m4n3 81m2n8256m1n2481m2n8256m1n24

111.

45p95q245p95q2 6442464424 128x852x25128x852x25

112.

80q55q80q55q −625353−625353 80m745m480m745m4

113.

50m72m50m72m 125023125023 486y92y34486y92y34

114.

72n112n72n112n 1626316263 160r105r34160r105r34

Writing Exercises

115.

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

116.

Explain why 7+97+9 is not equal to 7+9.7+9.

117.

Explain how you know that x105=x2.x105=x2.

118.

Explain why −644−644 is not a real number but −643−643 is.

Self Check

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

This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “use the product property to simplify radical expressions” and “use the quotient property to simplify radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

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

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