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Elementary Algebra 2e

9.4 Multiply Square Roots

Elementary Algebra 2e9.4 Multiply Square Roots
Table of contents
  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 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. 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 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. 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 in Two Variables
    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:

  • Multiply square roots
  • Use polynomial multiplication to multiply square roots

Be Prepared 9.9

Before you get started, take this readiness quiz.

Simplify: (3u)(8v)(3u)(8v).
 If you missed this problem, review Example 6.26.

Be Prepared 9.10

Simplify: 6(12−7n)6(12−7n).
 If you missed this problem, review Example 6.28.

Be Prepared 9.11

Simplify: (2+a)(4−a)(2+a)(4−a).
 If you missed this problem, review Example 6.39.

Multiply Square Roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says

ab=a·bab=a·b

We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.

a·b=aba·b=ab

Remember, we assume all variables are greater than or equal to zero.

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

Product Property of Square Roots

If a, b are nonnegative real numbers, then

ab=a·banda·b=abab=a·banda·b=ab

So we can multiply 3·53·5 in this way:

3·53·5153·53·515

Sometimes the product gives us a perfect square:

2·82·81642·82·8164

Even when the product is not a perfect square, we must look for perfect-square factors 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 12xy12xy. Keep this in mind as you do these examples.

Example 9.44

Simplify: ⓐ 2·62·6 ⓑ (43)(212)(43)(212).

Try It 9.87

Simplify: ⓐ 3·63·6 ⓑ (26)(312)(26)(312).

Try It 9.88

Simplify: ⓐ 5·105·10 ⓑ (63)(56)(63)(56).

Example 9.45

Simplify: (62)(310)(62)(310).

Try It 9.89

Simplify: (32)(230)(32)(230).

Try It 9.90

Simplify: (33)(36)(33)(36).

When we have to multiply square roots, we first find the product and then remove any perfect square factors.

Example 9.46

Simplify: ⓐ (8x3)(3x)(8x3)(3x) ⓑ (20y2)(5y3)(20y2)(5y3).

Try It 9.91

Simplify: ⓐ (6x3)(3x)(6x3)(3x) ⓑ (2y3)(50y2)(2y3)(50y2).

Try It 9.92

Simplify: ⓐ (6x5)(2x)(6x5)(2x) ⓑ (12y2)(3y5)(12y2)(3y5).

Example 9.47

Simplify: (106p3)(318p)(106p3)(318p).

Try It 9.93

Simplify: (62x2)(845x4)(62x2)(845x4).

Try It 9.94

Simplify: (26y4)(1230y)(26y4)(1230y).

Example 9.48

Simplify: ⓐ (2)2(2)2 ⓑ (−11)2(−11)2.

Try It 9.95

Simplify: ⓐ (12)2(12)2 ⓑ (−15)2(−15)2.

Try It 9.96

Simplify: ⓐ (16)2(16)2 ⓑ (−20)2(−20)2.

The results of the previous example lead us to this property.

Squaring a Square Root

If a is a nonnegative real number, then

(a)2=a(a)2=a

By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify (2)2(2)2 and get 2 right away. When we multiply the two like square roots in part (a) of the next example, it is the same as squaring.

Example 9.49

Simplify: ⓐ (23)(83)(23)(83) ⓑ (36)2(36)2.

Try It 9.97

Simplify: ⓐ (611)(511)(611)(511) ⓑ (58)2(58)2.

Try It 9.98

Simplify: ⓐ (37)(107)(37)(107) ⓑ (−46)2(−46)2.

Use Polynomial Multiplication to Multiply Square Roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.

We will first distribute and then simplify the square roots when possible.

Example 9.50

Simplify: ⓐ 3(5−2)3(5−2) ⓑ 2(4−10)2(4−10).

Try It 9.99

Simplify: ⓐ 2(3−5)2(3−5) ⓑ 3(2−18)3(2−18).

Try It 9.100

Simplify: ⓐ 6(2+6)6(2+6) ⓑ 7(1+14)7(1+14).

Example 9.51

Simplify: ⓐ 5(7+25)5(7+25) ⓑ 6(2+18)6(2+18).

Try It 9.101

Simplify: ⓐ 6(1+36)6(1+36) ⓑ 12(3+24)12(3+24).

Try It 9.102

Simplify: ⓐ 8(2−58)8(2−58) ⓑ 14(2+42)14(2+42).

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 9.52

Simplify: (2+3)(4−3)(2+3)(4−3).

Try It 9.103

Simplify: (1+6)(3−6)(1+6)(3−6).

Try It 9.104

Simplify: (4−10)(2+10)(4−10)(2+10).

Example 9.53

Simplify: (3−27)(4−27)(3−27)(4−27).

Try It 9.105

Simplify: (6−37)(3+47)(6−37)(3+47).

Try It 9.106

Simplify: (2−311)(4−11)(2−311)(4−11).

Example 9.54

Simplify: (32−5)(2+45)(32−5)(2+45).

Try It 9.107

Simplify: (53−7)(3+27)(53−7)(3+27).

Try It 9.108

Simplify: (6−38)(26+8)(6−38)(26+8)

Example 9.55

Simplify: (4−2x)(1+3x)(4−2x)(1+3x).

Try It 9.109

Simplify: (6−5m)(2+3m)(6−5m)(2+3m).

Try It 9.110

Simplify: (10+3n)(1−5n)(10+3n)(1−5n).

Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.

Special Product Formulas

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

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

Example 9.56

Simplify: ⓐ (2+3)2(2+3)2 ⓑ (4−25)2(4−25)2.

Try It 9.111

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

Try It 9.112

Simplify: ⓐ (6−5)2(6−5)2 ⓑ (9−210)2(9−210)2.

Example 9.57

Simplify: (1+3x)2(1+3x)2.

Try It 9.113

Simplify: (2+5m)2(2+5m)2.

Try It 9.114

Simplify: (3−4n)2(3−4n)2.

In the next two examples, we will find the product of conjugates.

Example 9.58

Simplify: (4−2)(4+2)(4−2)(4+2).

Try It 9.115

Simplify: (2−3)(2+3)(2−3)(2+3).

Try It 9.116

Simplify: (1+5)(1−5)(1+5)(1−5).

Example 9.59

Simplify: (5−23)(5+23)(5−23)(5+23).

Try It 9.117

Simplify: (3−25)(3+25)(3−25)(3+25).

Try It 9.118

Simplify: (4+57)(4−57)(4+57)(4−57).

Media

Access these online resources for additional instruction and practice with multiplying square roots.

Section 9.4 Exercises

Practice Makes Perfect

Multiply Square Roots

In the following exercises, simplify.

233.

ⓐ 2·82·8 ⓑ (33)(218)(33)(218)

234.

ⓐ 6·66·6 ⓑ (32)(232)(32)(232)

235.

ⓐ 7·147·14 ⓑ (48)(58)(48)(58)

236.

ⓐ 6·126·12 ⓑ (25)(210)(25)(210)

237.

( 5 2 ) ( 3 6 ) ( 5 2 ) ( 3 6 )

238.

( 2 3 ) ( 4 6 ) ( 2 3 ) ( 4 6 )

239.

( −2 3 ) ( 3 18 ) ( −2 3 ) ( 3 18 )

240.

( −4 5 ) ( 5 10 ) ( −4 5 ) ( 5 10 )

241.

( 5 6 ) ( − 12 ) ( 5 6 ) ( − 12 )

242.

( 6 2 ) ( − 10 ) ( 6 2 ) ( − 10 )

243.

( −2 7 ) ( −2 14 ) ( −2 7 ) ( −2 14 )

244.

( −2 11 ) ( −4 22 ) ( −2 11 ) ( −4 22 )

245.

ⓐ (15y)(5y3)(15y)(5y3) ⓑ (2n2)(18n3)(2n2)(18n3)

246.

ⓐ (14x3)(7x3)(14x3)(7x3) ⓑ (3q2)(48q3)(3q2)(48q3)

247.

ⓐ (16y2)(8y4)(16y2)(8y4) ⓑ (11s6)(11s)(11s6)(11s)

248.

ⓐ (8x3)(3x)(8x3)(3x) ⓑ (7r)(7r8)(7r)(7r8)

249.

( 2 5 b 3 ) ( 4 15 b ) ( 2 5 b 3 ) ( 4 15 b )

250.

( 3 8 c 5 ) ( 2 6 c 3 ) ( 3 8 c 5 ) ( 2 6 c 3 )

251.

( 5 2 d 7 ) ( 3 50 d 3 ) ( 5 2 d 7 ) ( 3 50 d 3 )

252.

4 6 t 2 3 3 t 2 4 6 t 2 3 3 t 2

253.

3 4 y 4 3 9 y 5 3 4 y 4 3 9 y 5

254.

( −2 7 z 3 ) ( 3 14 z 8 ) ( −2 7 z 3 ) ( 3 14 z 8 )

255.

( 4 2 k 5 ) ( −3 32 k 6 ) ( 4 2 k 5 ) ( −3 32 k 6 )

256.

ⓐ (7)2(7)2 ⓑ (−15)2(−15)2

257.

ⓐ (11)2(11)2 ⓑ (−21)2(−21)2

258.

ⓐ (19)2(19)2 ⓑ (−5)2(−5)2

259.


ⓐ (23)2(23)2
ⓑ (−3)2(−3)2

260.

ⓐ (411)(−311)(411)(−311) ⓑ (53)2(53)2

261.

ⓐ (213)(−913)(213)(−913) ⓑ (65)2(65)2

262.

ⓐ (−312)(−26)(−312)(−26) ⓑ (−410)2(−410)2

263.

ⓐ (−75)(−310)(−75)(−310) ⓑ (−214)2(−214)2

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

264.

ⓐ 3(4−3)3(4−3) ⓑ 2(4−6)2(4−6)

265.

ⓐ 4(6−11)4(6−11) ⓑ 2(5−12)2(5−12)

266.

ⓐ 5(3−7)5(3−7) ⓑ 3(4−15)3(4−15)

267.

ⓐ 7(−2−11)7(−2−11) ⓑ 7(6−14)7(6−14)

268.

ⓐ 7(5+27)7(5+27) ⓑ 5(10+18)5(10+18)

269.

ⓐ 11(8+411)11(8+411) ⓑ 3(12+27)3(12+27)

270.

ⓐ 11(−3+411)11(−3+411) ⓑ 3(15−18)3(15−18)

271.

ⓐ 2(−5+92)2(−5+92) ⓑ 7(3−21)7(3−21)

272.

( 8 + 3 ) ( 2 − 3 ) ( 8 + 3 ) ( 2 − 3 )

273.

( 7 + 3 ) ( 9 − 3 ) ( 7 + 3 ) ( 9 − 3 )

274.

( 8 − 2 ) ( 3 + 2 ) ( 8 − 2 ) ( 3 + 2 )

275.

( 9 − 2 ) ( 6 + 2 ) ( 9 − 2 ) ( 6 + 2 )

276.

( 3 − 7 ) ( 5 − 7 ) ( 3 − 7 ) ( 5 − 7 )

277.

( 5 − 7 ) ( 4 − 7 ) ( 5 − 7 ) ( 4 − 7 )

278.

( 1 + 3 10 ) ( 5 − 2 10 ) ( 1 + 3 10 ) ( 5 − 2 10 )

279.

( 7 − 2 5 ) ( 4 + 9 5 ) ( 7 − 2 5 ) ( 4 + 9 5 )

280.

( 3 + 10 ) ( 3 + 2 10 ) ( 3 + 10 ) ( 3 + 2 10 )

281.

( 11 + 5 ) ( 11 + 6 5 ) ( 11 + 5 ) ( 11 + 6 5 )

282.

( 2 7 − 5 11 ) ( 4 7 + 9 11 ) ( 2 7 − 5 11 ) ( 4 7 + 9 11 )

283.

( 4 6 + 7 13 ) ( 8 6 − 3 13 ) ( 4 6 + 7 13 ) ( 8 6 − 3 13 )

284.

( 5 − u ) ( 3 + u ) ( 5 − u ) ( 3 + u )

285.

( 9 − w ) ( 2 + w ) ( 9 − w ) ( 2 + w )

286.

( 7 + 2 m ) ( 4 + 9 m ) ( 7 + 2 m ) ( 4 + 9 m )

287.

( 6 + 5 n ) ( 11 + 3 n ) ( 6 + 5 n ) ( 11 + 3 n )

288.


ⓐ (3+5)2(3+5)2
ⓑ (2−53)2(2−53)2

289.

ⓐ (4+11)2(4+11)2 ⓑ (3−25)2(3−25)2

290.

ⓐ (9−6)2(9−6)2 ⓑ (10+37)2(10+37)2

291.

ⓐ (5−10)2(5−10)2 ⓑ (8+32)2(8+32)2

292.

( 3 − 5 ) ( 3 + 5 ) ( 3 − 5 ) ( 3 + 5 )

293.

( 10 − 3 ) ( 10 + 3 ) ( 10 − 3 ) ( 10 + 3 )

294.

( 4 + 2 ) ( 4 − 2 ) ( 4 + 2 ) ( 4 − 2 )

295.

( 7 + 10 ) ( 7 − 10 ) ( 7 + 10 ) ( 7 − 10 )

296.

( 4 + 9 3 ) ( 4 − 9 3 ) ( 4 + 9 3 ) ( 4 − 9 3 )

297.

( 1 + 8 2 ) ( 1 − 8 2 ) ( 1 + 8 2 ) ( 1 − 8 2 )

298.

( 12 − 5 5 ) ( 12 + 5 5 ) ( 12 − 5 5 ) ( 12 + 5 5 )

299.

( 9 − 4 3 ) ( 9 + 4 3 ) ( 9 − 4 3 ) ( 9 + 4 3 )

Mixed Practice

In the following exercises, simplify.

300.

3 · 21 3 · 21

301.

( 4 6 ) ( − 18 ) ( 4 6 ) ( − 18 )

302.

( −5 + 7 ) ( 6 + 21 ) ( −5 + 7 ) ( 6 + 21 )

303.

( −5 7 ) ( 6 21 ) ( −5 7 ) ( 6 21 )

304.

( −4 2 ) ( 2 18 ) ( −4 2 ) ( 2 18 )

305.

( 35 y 3 ) ( 7 y 3 ) ( 35 y 3 ) ( 7 y 3 )

306.

( 4 12 x 5 ) ( 2 6 x 3 ) ( 4 12 x 5 ) ( 2 6 x 3 )

307.

( 29 ) 2 ( 29 ) 2

308.

( −4 17 ) ( −3 17 ) ( −4 17 ) ( −3 17 )

309.

( −4 + 17 ) ( −3 + 17 ) ( −4 + 17 ) ( −3 + 17 )

Everyday Math

310.

A landscaper wants to put a square reflecting pool next to a triangular deck, as shown below. The triangular deck is a right triangle, with legs of length 9 feet and 11 feet, and the pool will be adjacent to the hypotenuse.

  1. ⓐ Use the Pythagorean Theorem to find the length of a side of the pool. Round your answer to the nearest tenth of a foot.
  2. ⓑ Find the exact area of the pool.
This figure is an illustration of a square pool with a deck in the shape of a right triangle. the pool's sides are x inches long while the deck's hypotenuse is x inches long and its legs are nine and eleven inches long.
311.

An artist wants to make a small monument in the shape of a square base topped by a right triangle, as shown below. The square base will be adjacent to one leg of the triangle. The other leg of the triangle will measure 2 feet and the hypotenuse will be 5 feet.

  1. ⓐ Use the Pythagorean Theorem to find the length of a side of the square base. Round your answer to the nearest tenth of a foot.
     This figure shows a marble sculpture in the form of a square with a right triangle resting on top of it. The sides of the square are x inches long, the legs of the triangle are x and two inches long, and the hypotenuse of the triangle is five inches long.
  2. ⓑ Find the exact area of the face of the square base.
312.

A square garden will be made with a stone border on one edge. If only 3+103+10 feet of stone are available, simplify (3+10)2(3+10)2 to determine the area of the largest such garden. Round your answer to the nearest tenth of a foot.

313.

A garden will be made so as to contain two square sections, one section with side length 5+65+6 yards and one section with side length 2+32+3 yards. Simplify 5+62+2+325+62+2+32 to determine the total area of the garden. Round your answer to the nearest tenth.

314.

Suppose a third section will be added to the garden in the previous exercise. The third section is to have a width of 432432 yards. Write an expression that gives the total area of the garden.

Writing Exercises

315.
  1. ⓐ Explain why (−n)2(−n)2 is always positive, for n≥0n≥0.
  2. ⓑ Explain why −n2−n2 is always negative, for n≥0n≥0.
316.

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 four columns and three rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no minus I don’t get it!” The rows under the “I can…” column read, “multiply square roots.,” and “use polynomial multiplication to multiply square roots.” The other rows under the other columns are empty.

ⓑ 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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