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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

Before you get started, take this readiness quiz.

  1. Simplify: 9 2 ( −9 ) 2 9 2 .
    If you missed this problem, review [link] .
  2. Round 3.846 to the nearest hundredth.
    If you missed this problem, review [link] .
  3. For each number, identify whether it is a real number or not a real number:
    100 −100 .
    If you missed this problem, review [link] .

Simplify expressions with square roots

Remember that when a number n is multiplied by itself, we write n 2 and read it “n squared.” For example, 15 2 reads as “15 squared,” and 225 is called the square of 15, since 15 2 = 225 .

Square of a number

If n 2 = m , then m is the square of n .

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is m is called a square root of m .

Square root of a number

If n 2 = m , then n is a square root of m .

Notice ( −15 ) 2 = 225 also, so −15 is also a square root of 225. Therefore, both 15 and −15 are square roots of 225.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, m , denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because 0 2 = 0 , 0 = 0 . Notice that zero has only one square root.

Square root notation

This figure is a picture of an m inside a square root sign. The sign is labeled as a radical sign and the m is labeled as the radicand.

m is read as “the square root of m .”

If m = n 2 , then m = n , for n 0 .

The square root of m , m , is the positive number whose square is m .

Since 15 is the positive square root of 225, we write 225 = 15 . Fill in [link] to make a table of square roots you can refer to as you work this chapter.

This table has fifteen columns and two rows. The first row contains the following numbers: the square root of 1, the square root of 4, the square root of 9, the square root of 16, the square root of 25, the square root of 36, the square root of 49, the square root of 64, the square root of 81, the square root of 100, the square root of 121, the square root of 144, the square root of 169, the square root of 196, and the square root of 225. The second row is completely empty except for the last column. The number 15 is in the last column.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 225 = 15 . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 225 = −15 .

Simplify: 36 196 81 289 .

Solution


36 Since 6 2 = 36 6


196 Since 14 2 = 196 14


81 The negative is in front of the radical sign. −9


289 The negative is in front of the radical sign. −17

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Simplify: 49 225 .

−7 15

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Simplify: 64 121 .

8 −11

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Simplify: −169 64 .

Solution


  1. −169 There is no real number whose square is −169 . −169 is not a real number.


  2. 64 The negative is in front of the radical. −8
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Simplify: −196 81 .

not a real number −9

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Simplify: 49 −121 .

−7 not a real number

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When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

Simplify: 25 + 144 25 + 144 .

Solution


25 + 144 Use the order of operations. 5 + 12 Simplify. 17


25 + 144 Simplify under the radical sign. 169 Simplify. 13

Notice the different answers in parts and !

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Simplify: 9 + 16 9 + 16 .

7 5

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Simplify: 64 + 225 64 + 225 .

17 23

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Estimate square roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at [link] below.

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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