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Elementary Algebra 2e

9.3 Add and Subtract Square Roots

Elementary Algebra 2e9.3 Add and Subtract Square Roots

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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:

  • Add and subtract like square roots
  • Add and subtract square roots that need simplification

Be Prepared 9.7

Before you get started, take this readiness quiz.

Add: 3x+9x3x+9x 5m+5n5m+5n.
If you missed this problem, review Example 1.24.

Be Prepared 9.8

Simplify: 50x350x3.
If you missed this problem, review Example 9.16.

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify 2+72+7 in this way:

2+7Add inside the radical.9Simplify.32+7Add inside the radical.9Simplify.3

So if we have to add 2+72+7, we must not combine them into one radical.

2+72+72+72+7

Trying to add square roots with different radicands is like trying to add unlike terms.

But, just like we can addx+x,we can add3+3.x+x=2x3+3=23But, just like we can addx+x,we can add3+3.x+x=2x3+3=23

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.

Like Square Roots

Square roots with the same radicand are called like square roots.

We add and subtract like square roots in the same way we add and subtract like terms. We know that 3x+8x3x+8x is 11x11x. Similarly we add 3x+8x3x+8x and the result is 11x.11x.

Add and Subtract Like Square Roots

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

Example 9.29

Simplify: 22722272.

Try It 9.57

Simplify: 82928292.

Try It 9.58

Simplify: 53935393.

Example 9.30

Simplify: 3y+4y3y+4y.

Try It 9.59

Simplify: 2x+7x2x+7x.

Try It 9.60

Simplify: 5u+3u5u+3u.

Example 9.31

Simplify: 4x2y4x2y.

Try It 9.61

Simplify: 7p6q7p6q.

Try It 9.62

Simplify: 6a3b6a3b.

Example 9.32

Simplify: 513+413+213513+413+213.

Try It 9.63

Simplify: 411+211+311411+211+311.

Try It 9.64

Simplify: 610+210+310610+210+310.

Example 9.33

Simplify: 2666+332666+33.

Try It 9.65

Simplify: 5545+265545+26.

Try It 9.66

Simplify: 3787+253787+25.

Example 9.34

Simplify: 25n65n+45n25n65n+45n.

Try It 9.67

Simplify: 7x77x+47x7x77x+47x.

Try It 9.68

Simplify: 43y73y+23y43y73y+23y.

When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

Example 9.35

Simplify: 3xy+53xy43xy3xy+53xy43xy.

Try It 9.69

Simplify: 5xy+45xy75xy5xy+45xy75xy.

Try It 9.70

Simplify: 37mn+7mn47mn37mn+7mn47mn.

Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

Example 9.36

Simplify: 20+3520+35.

Try It 9.71

Simplify: 18+6218+62.

Try It 9.72

Simplify: 27+4327+43.

Example 9.37

Simplify: 48754875.

Try It 9.73

Simplify: 32183218.

Try It 9.74

Simplify: 20452045.

Just like we use the Associative Property of Multiplication to simplify 5(3x)5(3x) and get 15x15x, we can simplify 5(3x)5(3x) and get 15x15x. We will use the Associative Property to do this in the next example.

Example 9.38

Simplify: 5182851828.

Try It 9.75

Simplify: 427312427312.

Try It 9.76

Simplify: 320745320745.

Example 9.39

Simplify: 34192561083419256108.

Try It 9.77

Simplify: 23108571472310857147.

Try It 9.78

Simplify: 35200341283520034128.

Example 9.40

Simplify: 2348341223483412.

Try It 9.79

Simplify: 25321382532138.

Try It 9.80

Simplify: 138014125138014125.

In the next example, we will remove constant and variable factors from the square roots.

Example 9.41

Simplify: 18n532n518n532n5.

Try It 9.81

Simplify: 32m750m732m750m7.

Try It 9.82

Simplify: 27p348p327p348p3.

Example 9.42

Simplify: 950m2648m2950m2648m2.

Try It 9.83

Simplify: 532x2348x2532x2348x2.

Try It 9.84

Simplify: 748y2472y2748y2472y2.

Example 9.43

Simplify: 28x25x32+518x228x25x32+518x2.

Try It 9.85

Simplify: 312x22x48+427x2312x22x48+427x2.

Try It 9.86

Simplify: 318x26x32+250x2318x26x32+250x2.

Media

Access this online resource for additional instruction and practice with the adding and subtracting square roots.

Section 9.3 Exercises

Practice Makes Perfect

Add and Subtract Like Square Roots

In the following exercises, simplify.

145.

8 2 5 2 8 2 5 2

146.

7 2 3 2 7 2 3 2

147.

3 5 + 6 5 3 5 + 6 5

148.

4 5 + 8 5 4 5 + 8 5

149.

9 7 10 7 9 7 10 7

150.

11 7 12 7 11 7 12 7

151.

7 y + 2 y 7 y + 2 y

152.

9 n + 3 n 9 n + 3 n

153.

a 4 a a 4 a

154.

b 6 b b 6 b

155.

5 c + 2 c 5 c + 2 c

156.

7 d + 2 d 7 d + 2 d

157.

8 a 2 b 8 a 2 b

158.

5 c 3 d 5 c 3 d

159.

5 m + n 5 m + n

160.

n + 3 p n + 3 p

161.

8 7 + 2 7 + 3 7 8 7 + 2 7 + 3 7

162.

6 5 + 3 5 + 5 6 5 + 3 5 + 5

163.

3 11 + 2 11 8 11 3 11 + 2 11 8 11

164.

2 15 + 5 15 9 15 2 15 + 5 15 9 15

165.

3 3 8 3 + 7 5 3 3 8 3 + 7 5

166.

5 7 8 7 + 6 3 5 7 8 7 + 6 3

167.

6 2 + 2 2 3 5 6 2 + 2 2 3 5

168.

7 5 + 5 8 10 7 5 + 5 8 10

169.

3 2 a 4 2 a + 5 2 a 3 2 a 4 2 a + 5 2 a

170.

11 b 5 11 b + 3 11 b 11 b 5 11 b + 3 11 b

171.

8 3 c + 2 3 c 9 3 c 8 3 c + 2 3 c 9 3 c

172.

3 5 d + 8 5 d 11 5 d 3 5 d + 8 5 d 11 5 d

173.

5 3 a b + 3 a b 2 3 a b 5 3 a b + 3 a b 2 3 a b

174.

8 11 c d + 5 11 c d 9 11 c d 8 11 c d + 5 11 c d 9 11 c d

175.

2 p q 5 p q + 4 p q 2 p q 5 p q + 4 p q

176.

11 2 r s 9 2 r s + 3 2 r s 11 2 r s 9 2 r s + 3 2 r s

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

177.

50 + 4 2 50 + 4 2

178.

48 + 2 3 48 + 2 3

179.

80 3 5 80 3 5

180.

28 4 7 28 4 7

181.

27 75 27 75

182.

72 98 72 98

183.

48 + 27 48 + 27

184.

45 + 80 45 + 80

185.

2 50 3 72 2 50 3 72

186.

3 98 128 3 98 128

187.

2 12 + 3 48 2 12 + 3 48

188.

4 75 + 2 108 4 75 + 2 108

189.

2 3 72 + 1 5 50 2 3 72 + 1 5 50

190.

2 5 75 + 3 4 48 2 5 75 + 3 4 48

191.

1 2 20 2 3 45 1 2 20 2 3 45

192.

2 3 54 3 4 96 2 3 54 3 4 96

193.

1 6 27 3 8 48 1 6 27 3 8 48

194.

1 8 32 1 10 50 1 8 32 1 10 50

195.

1 4 98 1 3 128 1 4 98 1 3 128

196.

1 3 24 + 1 4 54 1 3 24 + 1 4 54

197.

72 a 5 50 a 5 72 a 5 50 a 5

198.

48 b 5 75 b 5 48 b 5 75 b 5

199.

80 c 7 20 c 7 80 c 7 20 c 7

200.

96 d 9 24 d 9 96 d 9 24 d 9

201.

9 80 p 4 6 98 p 4 9 80 p 4 6 98 p 4

202.

8 72 q 6 3 75 q 6 8 72 q 6 3 75 q 6

203.

2 50 r 8 + 4 54 r 8 2 50 r 8 + 4 54 r 8

204.

5 27 s 6 + 2 20 s 6 5 27 s 6 + 2 20 s 6

205.

3 20 x 2 4 45 x 2 + 5 x 80 3 20 x 2 4 45 x 2 + 5 x 80

206.

2 28 x 2 63 x 2 + 6 x 7 2 28 x 2 63 x 2 + 6 x 7

207.

3 128 y 2 + 4 y 162 8 98 y 2 3 128 y 2 + 4 y 162 8 98 y 2

208.

3 75 y 2 + 8 y 48 300 y 2 3 75 y 2 + 8 y 48 300 y 2

Mixed Practice

209.

2 8 + 6 8 5 8 2 8 + 6 8 5 8

210.

2 3 27 + 3 4 48 2 3 27 + 3 4 48

211.

175 k 4 63 k 4 175 k 4 63 k 4

212.

5 6 162 + 3 16 128 5 6 162 + 3 16 128

213.

2 363 2 300 2 363 2 300

214.

150 + 4 6 150 + 4 6

215.

9 2 8 2 9 2 8 2

216.

5 x 8 y 5 x 8 y

217.

8 13 4 13 3 13 8 13 4 13 3 13

218.

5 12 c 4 3 27 c 6 5 12 c 4 3 27 c 6

219.

80 a 5 45 a 5 80 a 5 45 a 5

220.

3 5 75 1 4 48 3 5 75 1 4 48

221.

21 19 2 19 21 19 2 19

222.

500 + 405 500 + 405

223.

5 6 27 + 5 8 48 5 6 27 + 5 8 48

224.

11 11 10 11 11 11 10 11

225.

75 108 75 108

226.

2 98 4 72 2 98 4 72

227.

4 24 x 2 54 x 2 + 3 x 6 4 24 x 2 54 x 2 + 3 x 6

228.

8 80 y 6 6 48 y 6 8 80 y 6 6 48 y 6

Everyday Math

229.

A decorator decides to use square tiles as an accent strip in the design of a new shower, but she wants to rotate the tiles to look like diamonds. She will use 9 large tiles that measure 8 inches on a side and 8 small tiles that measure 2 inches on a side. Determine the width of the accent strip by simplifying the expression 9(82)+8(22)9(82)+8(22). (Round to the nearest tenth of an inch.)

230.

Suzy wants to use square tiles on the border of a spa she is installing in her backyard. She will use large tiles that have area of 12 square inches, medium tiles that have area of 8 square inches, and small tiles that have area of 4 square inches. Once section of the border will require 4 large tiles, 8 medium tiles, and 10 small tiles to cover the width of the wall. Simplify the expression 412+88+104412+88+104 to determine the width of the wall. (Round to the nearest tenth of an inch.)

Writing Exercises

231.

Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables.

232.

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.

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!” Under the “I can…” column the rows read, “add and subtract like square roots.,” and “add and subtract square roots that need simplification.” The other rows under the other 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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