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Elementary Algebra

9.3 Add and Subtract Square Roots

Elementary Algebra9.3 Add and Subtract Square 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 Solve 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 Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    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
    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:

  • Add and subtract like square roots
  • Add and subtract square roots that need simplification
Be Prepared 9.3

Before you get started, take this readiness quiz.

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

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.

82528252

146.

72327232

147.

35+6535+65

148.

45+8545+85

149.

9710797107

150.

117127117127

151.

7y+2y7y+2y

152.

9n+3n9n+3n

153.

a4aa4a

154.

b6bb6b

155.

5c+2c5c+2c

156.

7d+2d7d+2d

157.

8a2b8a2b

158.

5c3d5c3d

159.

5m+n5m+n

160.

n+3pn+3p

161.

87+27+3787+27+37

162.

65+35+565+35+5

163.

311+211811311+211811

164.

215+515915215+515915

165.

3383+753383+75

166.

5787+635787+63

167.

62+223562+2235

168.

75+581075+5810

169.

32a42a+52a32a42a+52a

170.

11b511b+311b11b511b+311b

171.

83c+23c93c83c+23c93c

172.

35d+85d115d35d+85d115d

173.

53ab+3ab23ab53ab+3ab23ab

174.

811cd+511cd911cd811cd+511cd911cd

175.

2pq5pq+4pq2pq5pq+4pq

176.

112rs92rs+32rs112rs92rs+32rs

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

177.

50+4250+42

178.

48+2348+23

179.

80358035

180.

28472847

181.

27752775

182.

72987298

183.

48+2748+27

184.

45+8045+80

185.

250372250372

186.

398128398128

187.

212+348212+348

188.

475+2108475+2108

189.

2372+15502372+1550

190.

2575+34482575+3448

191.

1220234512202345

192.

2354349623543496

193.

1627384816273848

194.

183211050183211050

195.

149813128149813128

196.

1324+14541324+1454

197.

72a550a572a550a5

198.

48b575b548b575b5

199.

80c720c780c720c7

200.

96d924d996d924d9

201.

980p4698p4980p4698p4

202.

872q6375q6872q6375q6

203.

250r8+454r8250r8+454r8

204.

527s6+220s6527s6+220s6

205.

320x2445x2+5x80320x2445x2+5x80

206.

228x263x2+6x7228x263x2+6x7

207.

3128y2+4y162898y23128y2+4y162898y2

208.

375y2+8y48300y2375y2+8y48300y2

Mixed Practice

209.

28+685828+6858

210.

2327+34482327+3448

211.

175k463k4175k463k4

212.

56162+31612856162+316128

213.

2363230023632300

214.

150+46150+46

215.

92829282

216.

5x8y5x8y

217.

813413313813413313

218.

512c4327c6512c4327c6

219.

80a545a580a545a5

220.

3575144835751448

221.

21192192119219

222.

500+405500+405

223.

5627+58485627+5848

224.

1111101111111011

225.

7510875108

226.

298472298472

227.

424x254x2+3x6424x254x2+3x6

228.

880y6648y6880y6648y6

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. 9(82)+8(22)9(82)+8(22). 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.

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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