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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

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
genevieve Reply
6.25
Ciid
6.25
Big
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Katherine Reply
complexed. could you please help me figure out?
Ciid
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
Marisol Reply
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
Mzo Reply
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
is anyone else having issues with the links not doing anything?
Helpful Reply
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
theresa Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form.
Samer Reply
Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at $4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use?
Samer
4x6.25= $25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds $77.96 they will spend in total
tyler
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
Sage Reply
i need help how to do this is confusing
Alvina Reply
what kind of math is it?
Danteii
help me to understand
Alvina Reply
huh, what is the algebra problem
Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
Christopher Reply
What is the domain and range of Heaviside and signum
Christopher
25-35
Fazal
The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Edi Reply
Tickets for a show are $70 for adults and $50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled $17,200. How many adult tickets and how many child tickets were sold?
Mum Reply
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
Edi Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?
Jesus Reply
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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