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Prealgebra

5.7 Simplify and Use Square Roots

Prealgebra5.7 Simplify and Use Square Roots
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. 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
  17. 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
  • Use square roots in applications
Be Prepared 5.7

Before you get started, take this readiness quiz.

  1. Simplify: (−9)2.(−9)2.
    If you missed this problem, review Example 3.52.
  2. Round 3.8463.846 to the nearest hundredth.
    If you missed this problem, review Example 5.9.
  3. Evaluate 12d12d for d=80.d=80.
    If you missed this problem, review Example 2.14.

Simplify Expressions with Square Roots

To start this section, we need to review some important vocabulary and notation.

Remember that when a number nn is multiplied by itself, we can write this as n2,n2, which we read aloud as nsquared.”nsquared.” For example, 8282 is read as “8squared.”“8squared.”

We call 6464 the square of 88 because 82=64.82=64. Similarly, 121121 is the square of 11,11, because 112=121.112=121.

Square of a Number

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

Modeling Squares

Do you know why we use the word square? If we construct a square with three tiles on each side, the total number of tiles would be nine.

A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.

This is why we say that the square of three is nine.

32=932=9

The number 99 is called a perfect square because it is the square of a whole number.

Manipulative Mathematics

Doing the Manipulative Mathematics activity Square Numbers will help you develop a better understanding of perfect square numbers

The chart shows the squares of the counting numbers 11 through 15.15. You can refer to it to help you identify the perfect squares.

A table with two columns is shown. The first column is labeled “Number” and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled “Square” and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

Perfect Squares

A perfect square is the square of a whole number.

What happens when you square a negative number?

(−8)2=(−8)(−8)=64(−8)2=(−8)(−8)=64

When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.

The chart shows the squares of the negative integers from −1−1 to −15.−15.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled “Square” and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

Did you notice that these squares are the same as the squares of the positive numbers?

Square Roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 102=100,102=100, we say 100100 is the square of 10.10. We can also say that 1010 is a square root of 100.100.

Square Root of a Number

A number whose square is mm is called a square root of m.m.

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

Notice (−10)2=100(−10)2=100 also, so −10−10 is also a square root of 100.100. Therefore, both 1010 and −10−10 are square roots of 100.100.

So, every positive number has two square roots: one positive and one negative.

What if we only want the positive square root of a positive number? The radical sign, 0,0, stands for the positive square root. The positive square root is also called the principal square root.

Square Root Notation

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

Ifm=n2,thenm=nforn0.Ifm=n2,thenm=nforn0.

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

We can also use the radical sign for the square root of zero. Because 02=0,0=0.02=0,0=0. Notice that zero has only one square root.

The chart shows the square roots of the first 1515 perfect square numbers.

A table is shown with 2 columns. The first column contains the values: square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second column contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.

Example 5.69

Simplify: 2525121.121.

Try It 5.137

Simplify: 3636169.169.

Try It 5.138

Simplify: 1616196.196.

Every positive number has two square roots and the radical sign indicates the positive one. We write 100=10.100=10. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 100=−10.100=−10.

Example 5.70

Simplify. 99144.144.

Try It 5.139

Simplify: 44225.225.

Try It 5.140

Simplify: 818164.64.

Square Root of a Negative Number

Can we simplify −25?−25? Is there a number whose square is −25?−25?

()2=−25?()2=−25?

None of the numbers that we have dealt with so far have a square that is −25.−25. Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to −25.−25. If we are asked to find the square root of any negative number, we say that the solution is not a real number.

Example 5.71

Simplify: −169−169121.121.

Try It 5.141

Simplify: −196−19681.81.

Try It 5.142

Simplify: −49−49121.121.

Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

Example 5.72

Simplify: 25+14425+14425+144.25+144.

Try It 5.143

Simplify: 9+169+169+16.9+16.

Try It 5.144

Simplify: 64+22564+22564+225.64+225.

Notice the different answers in parts and of Example 5.72. It is important to follow the order of operations correctly. In , we took each square root first and then added them. In , we added under the radical sign first and then found the square root.

Estimate Square Roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled “Square root” and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.

We might conclude that the square roots of numbers between 44 and 99 will be between 22 and 3,3, and they will not be whole numbers. Based on the pattern in the table above, we could say that 55 is between 22 and 3.3. Using inequality symbols, we write

2<5<32<5<3

Example 5.73

Estimate 6060 between two consecutive whole numbers.

Try It 5.145

Estimate 3838 between two consecutive whole numbers.

Try It 5.146

Estimate 8484 between two consecutive whole numbers.

Approximate Square Roots with a Calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the 00 or xx key on your calculator. You will to 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 number. It is an approximation, 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-digit10-digit display. Using it to find the square root of 55 will give 2.236067977.2.236067977. This is the approximate square root of 5.5. When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

52.23606797852.236067978

You will seldom use this many digits for applications in algebra. So, if you wanted to round 55 to two decimal places, you would write

52.2452.24

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

2.2360679782=5.0000000022.242=5.01762.2360679782=5.0000000022.242=5.0176

The squares are close, but not exactly equal, to 5.5.

Example 5.74

Round 1717 to two decimal places using a calculator.

Try It 5.147

Round 1111 to two decimal places.

Try It 5.148

Round 1313 to two decimal places.

Simplify Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Consider 9x2,9x2, where x0.x0. Can you think of an expression whose square is 9x2?9x2?

(?)2=9x2(3x)2=9x2so9x2=3x(?)2=9x2(3x)2=9x2so9x2=3x

When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

Example 5.75

Simplify: x2.x2.

Try It 5.149

Simplify: y2.y2.

Try It 5.150

Simplify: m2.m2.

Example 5.76

Simplify: 16x2.16x2.

Try It 5.151

Simplify: 64x2.64x2.

Try It 5.152

Simplify: 169y2.169y2.

Example 5.77

Simplify: 81y2.81y2.

Try It 5.153

Simplify: 121y2.121y2.

Try It 5.154

Simplify: 100p2.100p2.

Example 5.78

Simplify: 36x2y2.36x2y2.

Try It 5.155

Simplify: 100a2b2.100a2b2.

Try It 5.156

Simplify: 225m2n2.225m2n2.

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

How To

Use a strategy for applications with square roots.

  1. Step 1. Identify what you are asked to find.
  2. Step 2. Write a phrase that gives the information to find it.
  3. Step 3. Translate the phrase to an expression.
  4. Step 4. Simplify the expression.
  5. Step 5. Write a complete sentence that answers the question.

Square Roots and Area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.

If the area of the square is AA square units, the length of a side is AA units. See Table 5.7.

Area (square units) Length of side (units)
99 9=39=3
144144 144=12144=12
AA AA
Table 5.7

Example 5.79

Mike and Lychelle want to make a square patio. They have enough concrete for an area of 200200 square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

Try It 5.157

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370370 square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?

Try It 5.158

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 27042704 square centimeters. How long can a side of his mosaic be?

Square Roots and Gravity

Another application of square roots involves gravity. On Earth, if an object is dropped from a height of hh feet, the time in seconds it will take to reach the ground is found by evaluating the expression h4.h4. For example, if an object is dropped from a height of 6464 feet, we can find the time it takes to reach the ground by evaluating 644.644.

644644
Take the square root of 64. 8484
Simplify the fraction. 22

It would take 22 seconds for an object dropped from a height of 6464 feet to reach the ground.

Example 5.80

Christy dropped her sunglasses from a bridge 400400 feet above a river. How many seconds does it take for the sunglasses to reach the river?

Try It 5.159

A helicopter drops a rescue package from a height of 12961296 feet. How many seconds does it take for the package to reach the ground?

Try It 5.160

A window washer drops a squeegee from a platform 196196 feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?

Square Roots and Accident Investigations

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is dd feet, then the speed of the car can be found by evaluating 24d.24d.

Example 5.81

After a car accident, the skid marks for one car measured 190190 feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?

Try It 5.161

An accident investigator measured the skid marks of a car and found their length was 7676 feet. To the nearest tenth, what was the speed of the car before the brakes were applied?

Try It 5.162

The skid marks of a vehicle involved in an accident were 122122 feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?

Section 5.7 Exercises

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

489.

3636

490.

44

491.

6464

492.

144144

493.

44

494.

100100

495.

11

496.

121121

497.

−121−121

498.

−36−36

499.

−9−9

500.

−49−49

501.

9+169+16

502.

25+14425+144

503.

9+169+16

504.

25+14425+144

Estimate Square Roots

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

505.

7070

506.

5555

507.

200200

508.

172172

Approximate Square Roots with a Calculator

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

509.

1919

510.

2121

511.

5353

512.

4747

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

513.

y2y2

514.

b2b2

515.

49x249x2

516.

100y2100y2

517.

64a264a2

518.

25x225x2

519.

144x2y2144x2y2

520.

196a2b2196a2b2

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

521.

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 7575 square feet. How long can a side of his garden be?

522.

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130130 square feet. How long can a side of his patio be?

523.

Gravity An airplane dropped a flare from a height of 1,0241,024 feet above a lake. How many seconds did it take for the flare to reach the water?

524.

Gravity A hang glider dropped his cell phone from a height of 350350 feet. How many seconds did it take for the cell phone to reach the ground?

525.

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4,0004,000 feet above the Colorado River. How many seconds did it take for the hammer to reach the river?

526.

Accident investigation The skid marks from a car involved in an accident measured 5454 feet. What was the speed of the car before the brakes were applied?


527.

Accident investigation The skid marks from a car involved in an accident measured 216216 feet. What was the speed of the car before the brakes were applied?

528.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175175 feet. What was the speed of the vehicle before the brakes were applied?

529.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117117 feet. What was the speed of the vehicle before the brakes were applied?

Everyday Math

530.

Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy 625625 square centimeters of the designer tiles. How long can a side of the accent be?

531.

Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2,0252,025 tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?

Writing Exercises

532.

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

533.

What is the difference between 9292 and 9?9?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

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