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Elementary Algebra 2e

9.7 Higher Roots

Elementary Algebra 2e9.7 Higher 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. Key Terms
    13. Key Concepts
    14. 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. Key Terms
    10. Key Concepts
    11. 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. Key Terms
    9. Key Concepts
    10. 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. Key Terms
    10. Key Concepts
    11. 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 Solving 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. Key Terms
    9. Key Concepts
    10. 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. Key Terms
    10. Key Concepts
    11. 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 Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. 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. Key Terms
    12. Key Concepts
    13. 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. Key Terms
    11. Key Concepts
    12. 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 in Two Variables
    7. Key Terms
    8. Key Concepts
    9. 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:
  • Simplify expressions with higher roots
  • Use the Product Property to simplify expressions with higher roots
  • Use the Quotient Property to simplify expressions with higher roots
  • Add and subtract higher roots
Be Prepared 9.18

Before you get started, take this readiness quiz.

Simplify: y5y4y5y4.
If you missed this problem, review Example 6.18.

Be Prepared 9.19

Simplify: (n2)6(n2)6.
If you missed this problem, review Example 6.22.

Be Prepared 9.20

Simplify: x8x3x8x3.
If you missed this problem, review Example 6.59.

Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

We write:We say:n2nsquaredn3ncubedn4nto the fourthn5nto the fifthWe write:We say:n2nsquaredn3ncubedn4nto the fourthn5nto the fifth

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from −5to5−5to5. See Figure 9.4.

This figure consists of two tables. The first table shows the results of raising the numbers 1, 2, 3, 4, 5, x, and x squared to the second, third, fourth, and fifth powers. The second table shows the results of raising the numbers negative one through negative five to the second, third, fourth, and fifth powers. The table first has five columns and nine rows. The second has five columns and seven rows. The columns in both tables are labeled, “Number,” “Square,” “Cube,” “Fourth power,” “Fifth power,” nothing,  “Number,” “Square,” “Cube,” “Fourth power,” and “Fifth power.” In both tables, the next row reads: n, n squared, n cubed, n to the fourth power, n to the fifth power, nothing, n, n squared, n cubed, n to the fourth power, and n to the fifth power. In the first table, 1 squared, 1 cubed, 1 to the fourth power, and 1 to the fifth power are all shown to be 1. In the next row, 2 squared is 4, 2 cubed is 8, 2 to the fourth power is 16, and 2 to the fifth power is 32. In the next row, 3 squared is 9, 3 cubed is 27, 3 to the fourth power is 81, and 3 to the fifth power is 243. In the next row, 4 squared is 16, 4 cubed is 64, 4 to the fourth power is 246, and 4 to the fifth power is 1024. In the next row, 5 squared is 25, 5 cubed is 125, 5 to the fourth power is 625, and 5 to the fifth power is 3125. In the next row, x squared, x cubed, x to the fourth power, and x to the fifth power are listed. In the next row, x squared squared is x to the fourth power, x cubed squared is x to the fifth power, x squared to the fourth power is x to the eighth power, and x squared to the fifth power is x to the tenth power. In the second table, negative 1 squared is 1, negative 1 cubed is negative 1, negative 1 to the fourth power is 1, and negative 1 to the fifth power is negative 1. In the next row, negative 2 squared is 4, negative 2 cubed is negative 8, negative 2 to the fourth power is 16, and negative 2 to the fifth power is negative 32. In the next row, negative 4 squared is 16, negative 4 cubed is negative 64, negative 4 to the fourth power is 256, and negative 4 to the fifth power is negative 1024. In the next row, negative 5 squared is 25, negative 5 cubed is negative 125, negative 5 to the fourth power is 625, and negative 5 to the fifth power is negative 3125.
Figure 9.4 First through fifth powers of integers from −5−5 to 5.5.

Notice the signs in Figure 9.4. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2−2 below to help you see this.

This figure has five columns and two rows. The first row labels each column: n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row reads: negative 2, 4, negative 8, 16, and negative 32.

Earlier in this chapter we defined the square root of a number.

Ifn2=m,thennis a square root ofm.Ifn2=m,thennis a square root ofm.

And we have used the notation mm to denote the principal square root. So m0m0 always.

We will now extend the definition to higher roots.

nth Root of a Number

If bn=abn=a, then bb is an nth root of a number aa.

The principal nth root of aa is written anan.

        n is called the index of the radical.

We do not write the index for a square root. Just like we use the word ‘cubed’ for b3b3, we use the term ‘cube root’ for a3a3.

We refer to Figure 9.4 to help us find higher roots.

43=64643=434=81814=3(−2)5=−32−325=−243=64643=434=81814=3(−2)5=−32−325=−2

Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of anan

When nn is an even number and

  • a0a0, then anan is a real number
  • a<0a<0, then anan is not a real number

When nn is an odd number, anan is a real number for all values of aa.

Example 9.88

Simplify: 8383 814814 325325.

Try It 9.175

Simplify: 273273 25642564 24352435.

Try It 9.176

Simplify: 1000310003 164164 325325.

Example 9.89

Simplify: −643−643 −164−164 −2435−2435.

Try It 9.177

Simplify: −1253−1253 −164−164 −325−325.

Try It 9.178

Simplify: −2163−2163 −814−814 −10245−10245.

When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction.

The odd root of a number can be either positive or negative. We have seen that −643=−4−643=−4.

But the even root of a non-negative number is always non-negative, because we take the principal nth root.

Suppose we start with a=−5a=−5.

(−5)4=6256254=5(−5)4=6256254=5

How can we make sure the fourth root of −5 raised to the fourth power, (−5)4(−5)4 is 5? We will see in the following property.

Simplifying Odd and Even Roots

For any integer n2n2,

whennis oddann=awhennis evenann=|a|whennis oddann=awhennis evenann=|a|

We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Example 9.90

Simplify: x2x2 n33n33 p44p44 y55y55.

Try It 9.179

Simplify: b2b2 w33w33 m44m44 q55q55.

Try It 9.180

Simplify: y2y2 p33p33 z44z44 q55q55.

Example 9.91

Simplify: y183y183 z84z84.

Try It 9.181

Simplify: u124u124 v153v153.

Try It 9.182

Simplify: c205c205 d246d246.

Example 9.92

Simplify: 64p6364p63 16q12416q124.

Try It 9.183

Simplify: 27x27327x273 81q28481q284.

Try It 9.184

Simplify: 125p93125p93 243q255243q255.

Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of mnmn.

Simplified nth Root

anan is considered simplified if aa has no factors of mnmn.

We will generalize the Product Property of Square Roots to include any integer root n2n2.

Product Property of nth Roots

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

when anan and bnbn are real numbers and for any integer n2n2

Example 9.93

Simplify: x43x43 x74x74.

Try It 9.185

Simplify: y64y64 z53z53.

Try It 9.186

Simplify: p85p85 q136q136.

Example 9.94

Simplify: 163163 24342434.

Try It 9.187

Simplify: 813813 644644.

Try It 9.188

Simplify: 62536253 72947294.

Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Example 9.95

Simplify: 24x7324x73 80y14480y144.

Try It 9.189

Simplify: 54p10354p103 64q10464q104.

Try It 9.190

Simplify: 128m113128m113 162n74162n74.

Example 9.96

Simplify: −273−273 −164−164.

Try It 9.191

Simplify: −1083−1083 −484−484.

Try It 9.192

Simplify: −6253−6253 −3244−3244.

Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.

Example 9.97

Simplify: a8a53a8a53 a10a24a10a24.

Try It 9.193

Simplify: x7x34x7x34 y17y54y17y54.

Try It 9.194

Simplify: m13m73m13m73 n12n25n12n25.

Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.

Quotient Property of nth Roots

abn=anbnandanbn=abnabn=anbnandanbn=abn

when anandbnare real numbers,b0,and for any integern2anandbnare real numbers,b0,and for any integern2

Example 9.98

Simplify: −108323−108323 96x743x2496x743x24.

Try It 9.195

Simplify: −532323−532323 486m1143m54486m1143m54.

Try It 9.196

Simplify: −192333−192333 324n742n34324n742n34.

If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.

Example 9.99

Simplify: 24x7y3324x7y33 48x10y8448x10y84.

Try It 9.197

Simplify: 108c10d63108c10d63 80x10y5480x10y54.

Try It 9.198

Simplify: 40r3s340r3s3 162m14n124162m14n124.

Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.

Like Radicals

Radicals with the same index and same radicand are called like radicals.

Like radicals have the same index and the same radicand.

  • 942x4942x4 and −242x4−242x4 are like radicals.
  • 5125x35125x3 and 6125y36125y3 are not like radicals. The radicands are different.
  • 21000q521000q5 and −41000q4−41000q4 are not like radicals. The indices are different.

We add and subtract like radicals in the same way we add and subtract like terms. We can add 942x4+(−242x4)942x4+(−242x4) and the result is 742x4742x4.

Example 9.100

Simplify: 4x3+4x34x3+4x3 484284484284.

Try It 9.199

Simplify: 3x5+3x53x5+3x5 3939339393.

Try It 9.200

Simplify: 10y4+10y410y4+10y4 5326332653263326.

When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.

Example 9.101

Simplify: 543163543163 484+2434484+2434.

Try It 9.201

Simplify: 19238131923813 324+5124324+5124.

Try It 9.202

Simplify: 1283250312832503 645+4865645+4865.

Example 9.102

Simplify: 24x43−81x7324x43−81x73 162y94+516y54162y94+516y54.

Try It 9.203

Simplify: 32y53−108y8332y53−108y83 243r114+768r104243r114+768r104.

Try It 9.204

Simplify: 40z73−135z4340z73−135z43 80s134+1280s6480s134+1280s64.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with simplifying higher roots.

Section 9.7 Exercises

Practice Makes Perfect

Simplify Expressions with Higher Roots

In the following exercises, simplify.

442.

21632163 25642564 325325

443.

273273 164164 24352435

444.

51235123 814814 1515

445.

12531253 1296412964 1024510245

446.

−83−83 −814−814 −325−325

447.

−643−643 −164−164 −2435−2435

448.

−1253−1253 −12964−12964 −10245−10245

449.

−5123−5123 −814−814 −15−15

450.

u55u55 v88v88

451.
  1. a33a33

  2. .
452.

y44y44 m77m77

453.

k88k88 p66p66

454.

x93x93 y124y124

455.

a105a105 b273b273

456.

m84m84 n205n205

457.

r126r126 s303s303

458.

16x8416x84 64y12664y126

459.

−8c93−8c93 125d153125d153

460.

216a63216a63 32b20532b205

461.

128r147128r147 81s24481s244

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

462.

r53r53 s104s104

463.

u75u75 v116v116

464.

m54m54 n108n108

465.

p85p85 q83q83

466.

324324 647647

467.

62536253 12861286

468.

645645 25632563

469.

3125431254 813813

470.

108x53108x53 48y6448y64

471.

96a7596a75 375b43375b43

472.

405m104405m104 160n85160n85

473.

512p53512p53 324q74324q74

474.

−8643−8643 −2564−2564

475.

−4865−4865 −646−646

476.

−325−325 −18−18

477.

−83−83 −164−164

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

478.

p11p23p11p23 q17q134q17q134

479.

d12d75d12d75 m12m48m12m48

480.

u21u115u21u115 v30v126v30v126

481.

r14r53r14r53 c21c94c21c94

482.

6442464424 128x852x25128x852x25

483.

−625353−625353 80m745m480m745m4

484.

125023125023 486y92y34486y92y34

485.

1626316263 160r105r34160r105r34

486.

54a8b3354a8b33 64c5d2464c5d24

487.

96r11s3596r11s35 128u7v36128u7v36

488.

81s8t3381s8t33 64p15q12464p15q124

489.

625u10v33625u10v33 729c21d84729c21d84

Add and Subtract Higher Roots

In the following exercises, simplify.

490.

8p7+8p78p7+8p7 32532533253253

491.

15q3+15q315q3+15q3 2274627422746274

492.

39x5+79x539x5+79x5 83q723q783q723q7

493.

.

.
494.

81319238131923 51243245124324

495.

25035432503543 243418754243418754

496.

1283+25031283+2503 7295+9657295+965

497.

2434+125042434+12504 20003+54320003+543

498.

64a103−216a12364a103−216a123 486u74+768u34486u74+768u34

499.

80b53−270b3380b53−270b33 160v1041280v34160v1041280v34

Mixed Practice

In the following exercises, simplify.

500.

164164

501.

646646

502.

a33a33

503.
.
504.

−8c93−8c93

505.

125d153125d153

506.

r53r53

507.

s104s104

508.

108x53108x53

509.

48y6448y64

510.

−4865−4865

511.

−646−646

512.

6442464424

513.

128x852x25128x852x25

514.

96r11s3596r11s35

515.

128u7v36128u7v36

516.

81319238131923

517.

51243245124324

518.

64a103−216a12364a103−216a123

519.

486u74+768u34486u74+768u34

Everyday Math

520.

Population growth The expression 10·xn10·xn models the growth of a mold population after nn generations. There were 10 spores at the start, and each had xx offspring. So 10·xn10·xn is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression 10,24010510,240105 to determine the number of offspring of each spore.

521.

Spread of a virus The expression 3·xn3·xn models the spread of a virus after nn cycles. There were three people originally infected with the virus, and each of them infected xx people. So 3·x43·x4 is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression 187534187534 to determine the number of people each person infected.

Writing Exercises

522.

Explain how you know that x105=x2x105=x2 .

523.

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 four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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