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Intermediate Algebra

8.4 Add, Subtract, and Multiply Radical Expressions

Intermediate Algebra8.4 Add, Subtract, and Multiply 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:
  • Add and subtract radical expressions
  • Multiply radical expressions
  • Use polynomial multiplication to multiply radical expressions
Be Prepared 8.4

Before you get started, take this readiness quiz.

  1. Add: 3x2+9x5(x22x+3).3x2+9x5(x22x+3).
    If you missed this problem, review Example 5.5.
  2. Simplify: (2+a)(4a).(2+a)(4a).
    If you missed this problem, review Example 5.28.
  3. Simplify: (95y)2.(95y)2.
    If you missed this problem, review Example 5.31.

Add and Subtract Radical Expressions

Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.

Like Radicals

Like radicals are radical expressions with the same index and the same radicand.

We add and subtract like radicals in the same way we add and subtract like terms. We know that 3x+8x3x+8x is 11x.11x. Similarly we add 3x+8x3x+8x and the result is 11x.11x.

Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.

Example 8.36

Simplify: 22722272 5y3+4y35y3+4y3 7x42y4.7x42y4.

Try It 8.71

Simplify: 82928292 4x3+7x34x3+7x3 3x45y4.3x45y4.

Try It 8.72

Simplify: 53935393 5y3+3y35y3+3y3 5m42m3.5m42m3.

For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.

Example 8.37

Simplify: 25n65n+45n25n65n+45n 3xy4+53xy443xy4.3xy4+53xy443xy4.

Try It 8.73

Simplify: 7x77x+47x7x77x+47x 45xy4+25xy475xy4.45xy4+25xy475xy4.

Try It 8.74

Simplify: 43y73y+23y43y73y+23y 67mn3+7mn347mn3.67mn3+7mn347mn3.

Remember that we always simplify radicals by removing the largest factor from the radicand that is a power of the index. Once each radical is simplified, we can then decide if they are like radicals.

Example 8.38

Simplify: 20+3520+35 24337532433753 12484232434.12484232434.

Try It 8.75

Simplify: 18+6218+62 616322503616322503 2381312243.2381312243.

Try It 8.76

Simplify: 27+4327+43 45374034537403 12128353543.12128353543.

In the next example, we will remove both constant and variable factors from the radicals. Now that we have practiced taking both the even and odd roots of variables, it is common practice at this point for us to assume all variables are greater than or equal to zero so that absolute values are not needed. We will use this assumption throughout the rest of this chapter.

Example 8.39

Simplify: 950m2648m2950m2648m2 54n5316n53.54n5316n53.

Try It 8.77

Simplify: 32m750m732m750m7 135x7340x73.135x7340x73.

Try It 8.78

Simplify: 27p348p327p348p3 256y5332n53.256y5332n53.

Multiply Radical Expressions

We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of Roots ‘in reverse’ to multiply square roots. Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Roots so we see both ways together.

Product Property of Roots

For any real numbers, anan and bn,bn, and for any integer n2n2

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

When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply 4x·3y4x·3y we multiply the coefficients together and then the variables. The result is 12xy. Keep this in mind as you do these examples.

Example 8.40

Simplify: (62)(310)(62)(310) (−543)(−463).(−543)(−463).

Try It 8.79

Simplify: (32)(230)(32)(230) (2183)(−363).(2183)(−363).

Try It 8.80

Simplify: (33)(36)(33)(36) (−493)(363).(−493)(363).

We follow the same procedures when there are variables in the radicands.

Example 8.41

Simplify: (106p3)(43p)(106p3)(43p) (220y24)(328y34).(220y24)(328y34).

Try It 8.81

Simplify: (66x2)(830x4)(66x2)(830x4) (−412y34)(8y34).(−412y34)(8y34).

Try It 8.82

Simplify: (26y4)(1230y)(26y4)(1230y) (−49a34)(327a24).(−49a34)(327a24).

Use Polynomial Multiplication to Multiply Radical Expressions

In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals when possible.

Example 8.42

Simplify: 6(2+18)6(2+18) 93(5183).93(5183).

Try It 8.83

Simplify: 6(1+36)6(1+36) 43(−263).43(−263).

Try It 8.84

Simplify: 8(258)8(258) 33(9363).33(9363).

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

Example 8.43

Simplify: (327)(427)(327)(427) (x32)(x3+4).(x32)(x3+4).

Try It 8.85

Simplify: (637)(3+47)(637)(3+47) (x32)(x33).(x32)(x33).

Try It 8.86

Simplify: (2311)(411)(2311)(411) (x3+1)(x3+3).(x3+1)(x3+3).

Example 8.44

Simplify: (325)(2+45).(325)(2+45).

Try It 8.87

Simplify: (537)(3+27)(537)(3+27)

Try It 8.88

Simplify: (638)(26+8)(638)(26+8)

Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.

Special Products

Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(ab)=a2b2(ab)2=a22ab+b2Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(ab)=a2b2(ab)2=a22ab+b2

We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern.

Example 8.45

Simplify: (2+3)2(2+3)2 (425)2.(425)2.

Try It 8.89

Simplify: (10+2)2(10+2)2 (1+36)2.(1+36)2.

Try It 8.90

Simplify: (65)2(65)2 (9210)2.(9210)2.

In the next example, we will use the Product of Conjugates Pattern. Notice that the final product has no radical.

Example 8.46

Simplify: (523)(5+23).(523)(5+23).

Try It 8.91

Simplify: (325)(3+25)(325)(3+25)

Try It 8.92

Simplify: (4+57)(457).(4+57)(457).

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with adding, subtracting, and multiplying radical expressions.

Section 8.4 Exercises

Practice Makes Perfect

Add and Subtract Radical Expressions

In the following exercises, simplify.

165.

82528252 5m3+2m35m3+2m3 8m42m48m42m4

166.

72327232 7p3+2p37p3+2p3 5x33x35x33x3

167.

35+6535+65 9a3+3a39a3+3a3 52z4+2z452z4+2z4

168.

45+8545+85 m34m3m34m3 n+3nn+3n

169.

32a42a+52a32a42a+52a 53ab433ab423ab453ab433ab423ab4

170.

11b511b+311b11b511b+311b 811cd4+511cd4911cd4811cd4+511cd4911cd4

171.

83c+23c93c83c+23c93c 24pq354pq3+44pq324pq354pq3+44pq3

172.

35d+85d115d35d+85d115d 112rs392rs3+32rs3112rs392rs3+32rs3

173.

27752775 40332034033203 12324+23162412324+231624

174.

72987298 243+813243+813 1280423405412804234054

175.

48+2748+27 543+1283543+1283 6543280465432804

176.

45+8045+80 81319238131923 52804+73405452804+734054

177.

72a550a572a550a5 980p446405p44980p446405p44

178.

48b575b548b575b5 864q633125q63864q633125q63

179.

80c720c780c720c7 2162r104+432r1042162r104+432r104

180.

96d924d996d924d9 5243s64+23s645243s64+23s64

181.

3128y2+4y162898y23128y2+4y162898y2

182.

375y2+8y48300y2375y2+8y48300y2

Multiply Radical Expressions

In the following exercises, simplify.

183.

(−23)(318)(−23)(318) (843)(−4183)(843)(−4183)

184.

(−45)(510)(−45)(510) (−293)(793)(−293)(793)

185.

(56)(12)(56)(12) (−2184)(94)(−2184)(94)

186.

(−27)(−214)(−27)(−214) (−384)(−564)(−384)(−564)

187.

(412z3)(39z)(412z3)(39z) (53x33)(318x33)(53x33)(318x33)

188.

(32x3)(718x2)(32x3)(718x2) (−620a23)(−216a33)(−620a23)(−216a33)

189.

(−27z3)(314z8)(−27z3)(314z8) (28y24)(−212y34)(28y24)(−212y34)

190.

(42k5)(−332k6)(42k5)(−332k6) (6b34)(38b34)(6b34)(38b34)

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

191.

7(5+27)7(5+27) 63(4+183)63(4+183)

192.

11(8+411)11(8+411) 33(93+183)33(93+183)

193.

11(−3+411)11(−3+411) 34(544+184)34(544+184)

194.

2(−5+92)2(−5+92) 24(124+244)24(124+244)

195.

(7+3)(93)(7+3)(93)

196.

(82)(3+2)(82)(3+2)

197.

(932)(6+42)(932)(6+42) (x33)(x3+1)(x33)(x3+1)

198.

(327)(547)(327)(547) (x35)(x33)(x35)(x33)

199.

(1+310)(5210)(1+310)(5210) (2x3+6)(x3+1)(2x3+6)(x3+1)

200.

(725)(4+95)(725)(4+95) (3x3+2)(x32)(3x3+2)(x32)

201.

(3+10)(3+210)(3+10)(3+210)

202.

(11+5)(11+65)(11+5)(11+65)

203.

(27511)(47+911)(27511)(47+911)

204.

(46+713)(86313)(46+713)(86313)

205.

(3+5)2(3+5)2 (253)2(253)2

206.

(4+11)2(4+11)2 (325)2(325)2

207.

(96)2(96)2 (10+37)2(10+37)2

208.

(510)2(510)2 (8+32)2(8+32)2

209.

(4+2)(42)(4+2)(42)

210.

(7+10)(710)(7+10)(710)

211.

(4+93)(493)(4+93)(493)

212.

(1+82)(182)(1+82)(182)

213.

(1255)(12+55)(1255)(12+55)

214.

(943)(9+43)(943)(9+43)

215.

(3x3+2)(3x32)(3x3+2)(3x32)

216.

(4x3+3)(4x33)(4x3+3)(4x33)

Mixed Practice

217.

2327+34482327+3448

218.

175k463k4175k463k4

219.

56162+31612856162+316128

220.

243+/813243+/813

221.

1280423405412804234054

222.

813441343134813441343134

223.

512c4327c6512c4327c6

224.

80a545a580a545a5

225.

3575144835751448

226.

21932932193293

227.

864q633125q63864q633125q63

228.

1111101111111011

229.

3·213·21

230.

(46)(18)(46)(18)

231.

(743)(−3183)(743)(−3183)

232.

(412x5)(26x3)(412x5)(26x3)

233.

(29)2(29)2

234.

(−417)(−317)(−417)(−317)

235.

(−4+17)(−3+17)(−4+17)(−3+17)

236.

(38a24)(12a34)(38a24)(12a34)

237.

(632)2(632)2

238.

3(433)3(433)

239.

33(293+183)33(293+183)

240.

(6+3)(6+63)(6+3)(6+63)

Writing Exercises

241.

Explain the when a radical expression is in simplest form.

242.

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.

243.


Explain why (n)2(n)2 is always non-negative, for n0.n0.
Explain why (n)2(n)2 is always non-positive, for n0.n0.

244.

Use the binomial square pattern to simplify (3+2)2.(3+2)2. Explain all your steps.

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 “add and subtract radical expressions.”, “ multiply radical expressions”, and “use polynomial multiplication to multiply radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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