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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

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
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 televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
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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