5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (2024)

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 5.19

Before you get started, take this readiness quiz.

Simplify: (−9)2.(−9)2.
If you missed this problem, review Example 3.52.

Be Prepared 5.20

Round 3.8463.846 to the nearest hundredth.
If you missed this problem, review Example 5.9.

Be Prepared 5.21

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.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (1)

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.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (2)

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.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (3)

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.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (4)

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.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (5)

Example 5.69

Simplify: 2525 121.121.

Solution

2525
Since 52=2552=25 55
121121
Since 112=121112=121 1111

Try It 5.137

Simplify: 3636 169.169.

Try It 5.138

Simplify: 1616 196.196.

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. 99 144.144.

Solution

99
The negative is in front of the radical sign. 33
144144
The negative is in front of the radical sign. 1212

Try It 5.139

Simplify: 44 225.225.

Try It 5.140

Simplify: 8181 64.64.

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: −169−169 121.121.

Solution

There is no real number whose square is −169.−169. Therefore, −169−169 is not a real number.

The negative is in front of the radical sign, so we find the opposite of the square root of 121.121.

121121
The negative is in front of the radical. 1111

Try It 5.141

Simplify: −196−196 81.81.

Try It 5.142

Simplify: −49−49 121.121.

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+144 25+144.25+144.

Solution

Use the order of operations.
25+14425+144
Simplify each radical. 5+125+12
Add. 1717
Use the order of operations.
25+14425+144
Add under the radical sign. 169169
Simplify. 1313

Try It 5.143

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

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.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (6)

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.

Solution

Think of the perfect squares closest to 60.60. Make a small table of these perfect squares and their squares roots.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (7)

Locate 60 between two consecutive perfect squares.Locate 60 between two consecutive perfect squares.49<60<6449<60<64
60is between their square roots.60is between their square roots.7<60<87<60<8

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.

Solution

1717
Use the calculator square root key. 4.1231056264.123105626
Round to two decimal places. 4.124.12
174.12174.12

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.

Solution

Think about what we would have to square to get x2x2. Algebraically, (?)2=x2(?)2=x2

x2x2
Since (x)2=x2(x)2=x2 xx

Try It 5.149

Simplify: y2.y2.

Try It 5.150

Simplify: m2.m2.

Example 5.76

Simplify: 16x2.16x2.

Solution

16x216x2
Since(4x)2=16x2Since(4x)2=16x2 4x4x

Try It 5.151

Simplify: 64x2.64x2.

Try It 5.152

Simplify: 169y2.169y2.

Example 5.77

Simplify: 81y2.81y2.

Solution

81y281y2
Since(9y)2=81y2Since(9y)2=81y2 9y9y

Try It 5.153

Simplify: 121y2.121y2.

Try It 5.154

Simplify: 100p2.100p2.

Example 5.78

Simplify: 36x2y2.36x2y2.

Solution

36x2y236x2y2
Since(6xy)2=36x2y2Since(6xy)2=36x2y2 6xy6xy

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)
999=39=3
144144144=12144=12
AAAA

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?

Solution

We know the area of the square is 200200 square feet and want to find the length of the side. If the area of the square is AA square units, the length of a side is AA units.

What are you asked to find? The length of each side of a square patio
Write a phrase. The length of a side
Translate to an expression. AA
Evaluate AA when A=200A=200. 200200
Use your calculator. 14.142135...14.142135...
Round to one decimal place.14.1 feet14.1 feet
Write a sentence.Each side of the patio should be 14.114.1 feet.

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?

Solution

What are you asked to find? The number of seconds it takes for the sunglasses to reach the river
Write a phrase. The time it will take to reach the river
Translate to an expression. h4h4
Evaluate h4h4 when h=400h=400. 40044004
Find the square root of 400. 204204
Simplify.55
Write a sentence.It will take 5 seconds 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?

Solution

What are you asked to find? The speed of the car before the brakes were applied
Write a phrase. The speed of the car
Translate to an expression. 24d24d
Evaluate24d24dwhend=190.d=190. 24·19024·190
Multiply. 4,5604,560
Use your calculator.67.527772...67.527772...
Round to tenths.67.567.5
Write a sentence.The speed of the car was approximately 67.5 miles per hour.

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?

Links To Literacy

The Links to Literacy activity "Sea Squares" will provide you with another view of the topics covered in this section.

Media

ACCESS ADDITIONAL ONLINE RESOURCES

  • Introduction to Square Roots
  • Estimating Square Roots with a Calculator

Section 5.7 Exercises

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

489.

36 36

490.

4 4

491.

64 64

492.

144 144

493.

4 4

494.

100 100

495.

1 1

496.

121 121

497.

−121 −121

498.

−36 −36

499.

−9 −9

500.

−49 −49

501.

9 + 16 9 + 16

502.

25 + 144 25 + 144

503.

9 + 16 9 + 16

504.

25 + 144 25 + 144

Estimate Square Roots

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

505.

70 70

506.

55 55

507.

200 200

508.

172 172

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.

19 19

510.

21 21

511.

53 53

512.

47 47

Simplify Variable Expressions with Square Roots

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

513.

y 2 y 2

514.

b 2 b 2

515.

49 x 2 49 x 2

516.

100 y 2 100 y 2

517.

64 a 2 64 a 2

518.

25 x 2 25 x 2

519.

144 x 2 y 2 144 x 2 y 2

520.

196 a 2 b 2 196 a 2 b 2

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.

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (8)

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

5.7 Simplify and Use Square Roots - Prealgebra 2e | OpenStax (2024)

FAQs

How do you solve and simplify square roots? ›

Step 1: Find the prime factors of the number inside the radical sign. Step 2: Group the factors into pairs. Step 3: Pull out one integer outside the radical sign for each pair. Leave the other integers that could not be paired inside the radical sign.

How to simplify a square root with an exponent? ›

1) If the expression consists of a variable raised to an even power, the square root of the expression equals the variable raised to one-half of that power. Examples: 2) If the variable contains an odd power, express it as the product of two factors, one having an exponent 1 and the other with an even exponent.

What is a value of √5? ›

Therefore, the value of root 5 is, √5 = 2.2360… You can find the value of the square root of all the non-perfect square number with the help of the long division method.

What is the route of 2? ›

The square root of 2 or root 2 is represented using the square root symbol √ and written as √2 whose value is 1.414. This value is widely used in mathematics. Root 2 is an irrational number as it cannot be expressed as a fraction and has an infinite number of decimals.

How to calculate square root without a calculator? ›

To find the square root of a given square number by prime factorization, we follow the following steps:
  1. Obtain the prime factorization of the given natural number.
  2. Make pairs of identical factors.
  3. Take one factor from each pair and find their product. The product so obtained is the square root of the given number.

How to simplify the square root of 252? ›

The square root of 252 can be written as √252 and (252)1/2. The square root of 252 can be simplified to 6√7. The number 252 is not a perfect square as its square root is not an integer.

What does 5 √ 2 mean? ›

Thus, the square root of 50 is the value that is squared to get the original number. The simplified form of the square root of 50 is 5√2 or 7.07 (approximately). Square root of 50 can be represented in three forms. Radical form: √50 = 5√2.

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