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Elementary Algebra 2e

9.1 Simplify and Use Square Roots

Elementary Algebra 2e9.1 Simplify and Use 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 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 square roots
  • Estimate square roots
  • Approximate square roots
  • Simplify variable expressions with square roots
Be Prepared 9.1

Before you get started, take this readiness quiz.

Simplify: 9292 (−9)2(−9)2 9292.
If you missed this problem, review Example 1.50.

Be Prepared 9.2

Round 3.846 to the nearest hundredth.
If you missed this problem, review Example 1.94.

Be Prepared 9.3

For each number, identify whether it is a real number or not a real number:
100100 −100−100.
If you missed this problem, review Example 1.113.

Simplify Expressions with Square Roots

Remember that when a number nn is multiplied by itself, we write n2n2 and read it “n squared.” For example, 152152 reads as “15 squared,” and 225 is called the square of 15, since 152=225152=225.

Square of a Number

If n2=mn2=m, then mm is the square of nn.

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is mm is called a square root of mm.

Square Root of a Number

If n2=mn2=m, then nn is a square root of mm.

Notice (−15)2=225(−15)2=225 also, so −15−15 is also a square root of 225. Therefore, both 15 and −15−15 are square roots of 225.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, mm, denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because 02=002=0, 0=00=0. Notice that zero has only one square root.

Square Root Notation

This figure is a picture of an m inside a square root sign. The sign is labeled as a radical sign and the m is labeled as the radicand.

mm is read as “the square root of mm.”

If m=n2m=n2, then m=nm=n, for n0n0.

The square root of mm, mm, is the positive number whose square is mm.

Since 15 is the positive square root of 225, we write 225=15225=15. Fill in Figure 9.2 to make a table of square roots you can refer to as you work this chapter.

This table has fifteen columns and two rows. The first row contains the following numbers: the square root of 1, the square root of 4, the square root of 9, the square root of 16, the square root of 25, the square root of 36, the square root of 49, the square root of 64, the square root of 81, the square root of 100, the square root of 121, the square root of 144, the square root of 169, the square root of 196, and the square root of 225. The second row is completely empty except for the last column. The number 15 is in the last column.
Figure 9.2

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 225=15225=15. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 225=−15225=−15.

Example 9.1

Simplify: 3636 196196 8181 289289.

Try It 9.1

Simplify: 4949 225225.

Try It 9.2

Simplify: 6464 121121.

Example 9.2

Simplify: −169−169 6464.

Try It 9.3

Simplify: −196−196 8181.

Try It 9.4

Simplify: 4949 −121−121.

When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

Example 9.3

Simplify: 25+14425+144 25+14425+144.

Try It 9.5

Simplify: 9+169+16 9+169+16.

Try It 9.6

Simplify: 64+22564+225 64+22564+225.

Estimate Square Roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at Table 9.1 below.

Number Square Root
4 44 = 2
5 55
6 66
7 77
8 88
9 99 = 3
Table 9.1

The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that 55 must be between 2 and 3. Using inequality symbols, we write:

2<5<32<5<3

Example 9.4

Estimate 6060 between two consecutive whole numbers.

Try It 9.7

Estimate the square root 3838 between two consecutive whole numbers.

Try It 9.8

Estimate the square root 8484 between two consecutive whole numbers.

Approximate Square Roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the xx key on your calculator. You will use this key to approximate square roots.

When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is and it is read ‘approximately.’

Suppose your calculator has a 10-digit display. You would see that

52.23606797852.236067978

If we wanted to round 55 to two decimal places, we would say

52.2452.24

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

(2.236067978)2=5.000000002(2.24)2=5.0176(2.236067978)2=5.000000002(2.24)2=5.0176

Their squares are close to 5, but are not exactly equal to 5.

Using the square root key on a calculator and then rounding to two decimal places, we can find:

4=252.2462.4572.6582.839=34=252.2462.4572.6582.839=3

Example 9.5

Round 1717 to two decimal places.

Try It 9.9

Round 1111 to two decimal places.

Try It 9.10

Round 1313 to two decimal places.

Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider 9x29x2. Can you think of an expression whose square is 9x29x2?

(?)2=9x2(3x)2=9x2,so9x2=3x(?)2=9x2(3x)2=9x2,so9x2=3x

When we use the radical sign to take the square root of a variable expression, we should specify that x0x0 to make sure we get the principal square root.

However, in this chapter we will assume that each variable in a square-root expression represents a non-negative number and so we will not write x0x0 next to every radical.

What about square roots of higher powers of variables? Think about the Power Property of Exponents we used in Chapter 6.

(am)n=am·n(am)n=am·n

If we square amam, the exponent will become 2m2m.

(am)2=a2m(am)2=a2m

How does this help us take square roots? Let’s look at a few:

25u8=5u4because(5u4)2=25u816r20=4r10because(4r10)2=16r20196q36=14q18because(14q18)2=196q3625u8=5u4because(5u4)2=25u816r20=4r10because(4r10)2=16r20196q36=14q18because(14q18)2=196q36

Example 9.6

Simplify: x6x6 y16y16.

Try It 9.11

Simplify: y8y8 z12z12.

Try It 9.12

Simplify: m4m4 b10b10.

Example 9.7

Simplify: 16n216n2.

Try It 9.13

Simplify: 64x264x2.

Try It 9.14

Simplify: 169y2169y2.

Example 9.8

Simplify: 81c281c2.

Try It 9.15

Simplify: 121y2121y2.

Try It 9.16

Simplify: 100p2100p2.

Example 9.9

Simplify: 36x2y236x2y2.

Try It 9.17

Simplify: 100a2b2100a2b2.

Try It 9.18

Simplify: 225m2n2225m2n2.

Example 9.10

Simplify: 64p6464p64.

Try It 9.19

Simplify: 49x3049x30.

Try It 9.20

Simplify: 81w3681w36.

Example 9.11

Simplify: 121a6b8121a6b8

Try It 9.21

Simplify: 169x10y14169x10y14.

Try It 9.22

Simplify: 144p12q20144p12q20.

Media Access Additional Online Resources

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

Section 9.1 Exercises

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

1.

3636

2.

44

3.

6464

4.

169169

5.

99

6.

1616

7.

100100

8.

144144

9.

44

10.

100100

11.

11

12.

121121

13.

−121−121

14.

−36−36

15.

−9−9

16.

−49−49

17.

9+169+16

18.

25+14425+144

19.

9+169+16

20.

25+14425+144

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

21.

7070

22.

5555

23.

200200

24.

172172

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

25.

1919

26.

2121

27.

5353

28.

4747

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

29.

y2y2

30.

b2b2

31.

a14a14

32.

w24w24

33.

49x249x2

34.

100y2100y2

35.

121m20121m20

36.

25h4425h44

37.

81x3681x36

38.

144z84144z84

39.

81x1881x18

40.

100m32100m32

41.

64a264a2

42.

25x225x2

43.

144x2y2144x2y2

44.

196a2b2196a2b2

45.

169w8y10169w8y10

46.

81p24q681p24q6

47.

9c8d129c8d12

48.

36r6s2036r6s20

Everyday Math

49.

Decorating Denise wants to have a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?

50.

Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2025 square inch tiles. How long can a side of the mosaic be?

Writing Exercises

51.

Why is there no real number equal to −64−64?

52.

What is the difference between 9292 and 99?

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 columns are labeled, “I can…,” “Confidentally,” “With some help,” and “No – I don’t get it!” Under the “I can…,” column are, “simplify expressions with square roots.,” “estimate square roots.,” “approximate square roots.,” and “4) simplify variable expressions with square roots.” All the other rows under the different 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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