Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

9.1: Simplify and Use Square Roots

( \newcommand{\kernel}{\mathrm{null}\,}\)

Learning Objectives

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
BE PREPARED

Before you get started, take this readiness quiz.

  1. Simplify: ⓐ 92(9)292.
    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:
    100100.
    If you missed this problem, review [link].

Simplify Expressions with Square Roots

Remember that when a number n is multiplied by itself, we write n2 and read it “n squared.” For example, 152 reads as “15 squared,” and 225 is called the square of 15, since 152=225.

Definition: SQUARE OF A NUMBER

If n2=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.

Definition: SQUARE ROOT OF A NUMBER

If n2=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 02=0, 0=0. Notice that zero has only one square root.

Definition: 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=n2, then m=n, for n0.

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 Figure 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.

Example 9.1.1

Simplify:

  1. 36
  2. 196
  3. 81
  4. 289.
Answer

1.
36Since62=366
2.
196Since142=19614
3.
81The negative is in front of the radical sign9
4.
289The negative is in front of the radical sign17

Example 9.1.2

Simplify:

  1. 49
  2. 225.
Answer
  1. −7
  2. 15
Example 9.1.3

implify:

  1. 64
  2. 121.
Answer
  1. 8
  2. −11
Example 9.1.4

Simplify:

  1. 169
  2. 64
Answer

1.

169There is no real number whose square iss169169is not a real number.

2.

64The negative is in front of the radical sign8

Example 9.1.5

Simplify:

  1. 196
  2. 81.
Answer
  1. not a real number
  2. −9
Example 9.1.6

Simplify:

  1. 49
  2. 121.
Answer
  1. −7
  2. not a real number

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

Example 9.1.7

Simplify:

  1. 25+144
  2. 25+144.
Answer

1.

25+144Use the order of operations5+12Simplify.17

2.

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

Notice the different answers in parts 1 and 2!

Example 9.1.8

Simplify:

  1. 9+16
  2. 9+16.
Answer
  1. 7
  2. 5
Example 9.1.9

Simplify:

  1. 64+225
  2. 64+225.
Answer
  1. 17
  2. 23

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 Table below.

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

Example 9.1.10

Estimate 60 between two consecutive whole numbers.

Answer

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. .
Example 9.1.11

Estimate the square root 38 between two consecutive whole numbers.

Answer

6<38<7

Example 9.1.12

Estimate the square root 84 between two consecutive whole numbers.

Answer

9<84<10

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

52.236067978

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

52.24

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=252.2462.4572.6582.839=3

Example 9.1.13

Round 17 to two decimal places.

Answer

17Use the calculator square root key.4.123105626...Round to two decimal places.4.12174.12

Example 9.1.14

Round 11 to two decimal places.

Answer

3.32

Example 9.1.15

Round 13 to two decimal places.

Answer

3.61

Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider 9x2. Can you think of an expression whose square is 9x2?

(?)2=9x2(3x)2=9x2so9x2=3x

When we use the radical sign to take the square root of a variable expression, we should specify that x≥0x≥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 x0 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.

(am)n=am·n

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

(am)2=a2m

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

25u8=5u4Because(5u4)2=25u816r20=4r10Because(4r10)2=16r20196q36=14q18Because(14r18)2=196q36

Example 9.1.16

Simplify:

  1. x6
  2. y16
Answer

1.

x6Since(x3)2=x6x3

2.

y16Since(y8)2=y16y8

Example 9.1.17

Simplify:

  1. y8
  2. z12.
Answer
  1. y4
  2. z6
Example 9.1.18

Simplify:

  1. m4
  2. b10.
Answer
  1. m2
  2. b5
Example 9.1.19

Simplify: 16n2

Answer

16n2Since(4n)2=16n24n

Example 9.1.20

Simplify: 64x2.

Answer

8x

Example 9.1.21

Simplify: 169y2.

Answer

13y

Example 9.1.22

Simplify: 81c2.

Answer

81c2Since(9c)2=81c29c

Example 9.1.23

Simplify: 121y2.

Answer

11y

Example 9.1.24

Simplify: 100p2.

Answer

10p

Example 9.1.25

Simplify: 36x2y2.

Answer

36x2y2Since(6xy)2=36x2y26xy

Example 9.1.26

Simplify: 100a2b2.

Answer

10ab

Example 9.1.27

Simplify: 225m2n2.

Answer

15mn

Example 9.1.28

Simplify: 64p64.

Answer

64p64Since(8p8)2=64p648p8

Example 9.1.29

Simplify: 49x30.

Answer

7x15

Example 9.1.30

Simplify: 81w36

Answer

9w18

Example 9.1.31

Simplify: 121a6b8

Answer

121a6b8Since(11a3b4)2=121a6b811a3b4

Example 9.1.32

Simplify: 169x10y14

Answer

13x5y7

Example 9.1.33

Simplify: 144p12q20

Answer

12p6q10

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

  • 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 x0 to make sure we get the principal square root.

Glossary

square of a number
  • If n2=m, then m is the square of n
square root of a number
  • If n2=m, then n is a square root of m
square root notation
  • If m=n2, then m=n. We read m as ‘the square root of m.’

This page titled 9.1: Simplify and Use Square Roots is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by OpenStax.

Support Center

How can we help?