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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

how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply
What is observation
adeyemi Reply
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
Sunnyshay Reply
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Sophia Reply
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs $9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot?
Parker Reply
what you wanna get
Cesar
800 sq. ft @ $1.50 & 200 sq. ft @ $9.00
Marco
Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
Enith Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Alexandra Reply
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Cecilia Reply
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year
Heidi Reply
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Elbert Reply
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
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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