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Number Square Root
4 4 = 2
5 5
6 6
7 7
8 8
9 9 = 3

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 5 must be between 2 and 3. Using inequality symbols, we write:

2 < 5 < 3

Estimate 60 between two consecutive whole numbers.

Solution

Think of the perfect square numbers closest to 60. Make a small table of these perfect squares and their squares roots.

.
Locate 60 between two consecutive perfect squares. .
60 is between their square roots. .
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Estimate the square root 38 between two consecutive whole numbers.

6 < 38 < 7

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Estimate the square root 84 between two consecutive whole numbers.

9 < 84 < 10

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Approximate square roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the x 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

5 2.236067978

If we wanted to round 5 to two decimal places, we would say

5 2.24

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

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 = 2 5 2.24 6 2.45 7 2.65 8 2.83 9 = 3

Round 17 to two decimal places.

Solution

17 Use the calculator square root key. 4.123105626... Round to two decimal places. 4.12 17 4.12

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Round 11 to two decimal places.

3.32

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Round 13 to two decimal places.

3.61

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Simplify variable expressions with square roots

What if we have to find a square root of an expression with a variable? Consider 9 x 2 . Can you think of an expression whose square is 9 x 2 ?

( ? ) 2 = 9 x 2 ( 3 x ) 2 = 9 x 2 , so 9 x 2 = 3 x

When we use the radical sign to take the square root of a variable expression, we should specify that x 0 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 x 0 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.

( a m ) n = a m · n

If we square a m , the exponent will become 2 m .

( a m ) 2 = a 2 m

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

25 u 8 = 5 u 4 because ( 5 u 4 ) 2 = 25 u 8 16 r 20 = 4 r 10 because ( 4 r 10 ) 2 = 16 r 20 196 q 36 = 14 q 18 because ( 14 q 18 ) 2 = 196 q 36

Simplify: x 6 y 16 .

Solution


x 6 Since ( x 3 ) 2 = x 6 . x 3


y 16 Since ( y 8 ) 2 = y 16 . y 8

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Simplify: y 8 z 12 .

y 4 z 6

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Simplify: m 4 b 10 .

m 2 b 5

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Simplify: 16 n 2 .

Solution

16 n 2 Since ( 4 n ) 2 = 16 n 2 . 4 n

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Simplify: 169 y 2 .

13 y

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Simplify: 81 c 2 .

Solution

81 c 2 Since ( 9 c ) 2 = 81 c 2 . −9 c

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Simplify: 121 y 2 .

−11 y

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Simplify: 100 p 2 .

−10 p

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Simplify: 36 x 2 y 2 .

Solution

36 x 2 y 2 Since ( 6 x y ) 2 = 36 x 2 y 2 . 6 x y

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Simplify: 100 a 2 b 2 .

10 a b

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Simplify: 225 m 2 n 2 .

15 m n

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Simplify: 64 p 64 .

Solution

64 p 64 Since ( 8 p 32 ) 2 = 64 p 64 . 8 p 32

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Simplify: 49 x 30 .

7 x 15

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Simplify: 81 w 36 .

9 w 18

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Simplify: 121 a 6 b 8

Solution

121 a 6 b 8 Since ( 11 a 3 b 4 ) 2 = 121 a 6 b 8 . 11 a 3 b 4

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Simplify: 169 x 10 y 14 .

13 x 5 y 7

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Simplify: 144 p 12 q 20 .

12 p 6 q 10

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

Key concepts

  • Note that the square root of a negative number is not a real number.
  • Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.
  • We can estimate square roots using nearby perfect squares.
  • We can approximate square roots using a calculator.
  • When we use the radical sign to take the square root of a variable expression, we should specify that x 0 to make sure we get the principal square root.

Practice makes perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

−121

not a real number

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Estimate Square Roots

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

Approximate Square Roots

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

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Everyday math

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?

25 centimeters

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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?

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Writing exercises

Why is there no real number equal to −64 ?

Answers will vary.

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What is the difference between 9 2 and 9 ?

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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?

Questions & Answers

a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910.
David Reply
. A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Mikaela Reply
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Sam Reply
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Mckenzie Reply
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?
Jake Reply
enrique borrowed $23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total $23,500
Nakiya Reply
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Bridget Reply
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
help me understand graphs
Marlene Reply
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
what's the formula given? f(x)=?
Melissa
It shows a graph that I wish I could send photo of to you on here
Marlene
Which problem specifically?
Melissa
which problem?
Melissa
I don't know any to be honest. But whatever you can help me with for I can practice will help
Marlene
I got it. sorry, was out and about. I'll look at it now.
Melissa
Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things.
Marlene
I just responded.
Melissa
Thank you
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
WENDY Reply
State the question clearly please
Rich
write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
0.81818
Rich
5/100 = .05 but Rich is right that 9/11 = .81818
Melissa
Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
write in simplest form 3 4/2
August
Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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