Roots and Radicals

72 Simplify and Use Square Roots

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

Before you get started, take this readiness quiz.

  1. Simplify: {9}^{2} {\left(-9\right)}^{2} \text{−}{9}^{2}.
    If you missed this problem, review (Figure).
  2. Round 3.846 to the nearest hundredth.
    If you missed this problem, review (Figure).
  3. For each number, identify whether it is a real number or not a real number:
    \text{−}\sqrt{100} \sqrt{-100}.
    If you missed this problem, review (Figure).

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 {\left(-15\right)}^{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, \sqrt{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, \sqrt{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.

\sqrt{m} is read as “the square root of m.”

If m={n}^{2}, then \sqrt{m}=n, for n\ge 0.

The square root of m, \sqrt{m}, is the positive number whose square is m.

Since 15 is the positive square root of 225, we write \sqrt{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 \sqrt{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, \text{−}\sqrt{225}=-15.

Simplify: \sqrt{36} \sqrt{196} \text{−}\sqrt{81} \text{−}\sqrt{289}.

Solution


\begin{array}{cccc}& & & \phantom{\rule{11em}{0ex}}\sqrt{36}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{6}^{2}=36\hfill & & & \phantom{\rule{11em}{0ex}}6\hfill \end{array}


\begin{array}{cccc}& & & \phantom{\rule{10.5em}{0ex}}\sqrt{196}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{14}^{2}=196\hfill & & & \phantom{\rule{10.5em}{0ex}}14\hfill \end{array}


\begin{array}{cccc}& & & \text{−}\sqrt{81}\hfill \\ \text{The negative is in front of the radical sign.}\hfill & & & -9\hfill \end{array}


\begin{array}{cccc}& & & \text{−}\sqrt{289}\hfill \\ \text{The negative is in front of the radical sign.}\hfill & & & -17\hfill \end{array}

Simplify: \text{−}\sqrt{49} \sqrt{225}.

-715

Simplify: \sqrt{64} \text{−}\sqrt{121}.

8-11

Simplify: \sqrt{-169} \text{−}\sqrt{64}.

Solution

  1. \begin{array}{cccc}& & & \sqrt{-169}\hfill \\ \\ \\ \text{There is no real number whose square is}\phantom{\rule{0.2em}{0ex}}-169.\hfill & & & \sqrt{-169}\phantom{\rule{0.2em}{0ex}}\text{is not a real number.}\hfill \end{array}


  2. \begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\text{−}\sqrt{64}\hfill \\ \\ \\ \text{The negative is in front of the radical.}\hfill & & & \phantom{\rule{4em}{0ex}}-8\hfill \end{array}

Simplify: \sqrt{-196} \text{−}\sqrt{81}.

not a real number -9

Simplify: \text{−}\sqrt{49} \sqrt{-121}.

-7 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.

Simplify: \sqrt{25}+\sqrt{144} \sqrt{25+144}.

Solution


\begin{array}{cccc}& & & \phantom{\rule{3em}{0ex}}\sqrt{25}+\sqrt{144}\hfill \\ \\ \\ \text{Use the order of operations.}\hfill & & & \phantom{\rule{3em}{0ex}}5+12\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \phantom{\rule{3em}{0ex}}17\hfill \end{array}


\begin{array}{cccc}& & & \phantom{\rule{2em}{0ex}}\sqrt{25+144}\hfill \\ \\ \\ \text{Simplify under the radical sign.}\hfill & & & \phantom{\rule{2em}{0ex}}\sqrt{169}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \phantom{\rule{2em}{0ex}}13\hfill \end{array}

Notice the different answers in parts and !

Simplify: \sqrt{9}+\sqrt{16} \sqrt{9+16}.

7 5

Simplify: \sqrt{64+225} \sqrt{64}+\sqrt{225}.

17 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 (Figure) below.

Number Square Root
4 \sqrt{4} = 2
5 \sqrt{5}
6 \sqrt{6}
7 \sqrt{7}
8 \sqrt{8}
9 \sqrt{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 \sqrt{5} must be between 2 and 3. Using inequality symbols, we write:

2<\sqrt{5}<3

Estimate \sqrt{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. .
\sqrt{60} is between their square roots. .

Estimate the square root \sqrt{38} between two consecutive whole numbers.

6<\sqrt{38}<7

Estimate the square root \sqrt{84} between two consecutive whole numbers.

9<\sqrt{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 \sqrt{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 \approx and it is read ‘approximately.’

Suppose your calculator has a 10-digit display. You would see that

\phantom{\rule{3.4em}{0ex}}\sqrt{5}\approx 2.236067978

If we wanted to round \sqrt{5} to two decimal places, we would say

\sqrt{5}\approx 2.24

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

\begin{array}{ccc}\hfill {\left(2.236067978\right)}^{2}& =\hfill & 5.000000002\hfill \\ \hfill {\left(2.24\right)}^{2}& =\hfill & 5.0176\hfill \end{array}

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:

\begin{array}{ccc}\hfill \sqrt{4}& =\hfill & 2\hfill \\ \hfill \sqrt{5}& \approx \hfill & 2.24\hfill \\ \hfill \sqrt{6}& \approx \hfill & 2.45\hfill \\ \hfill \sqrt{7}& \approx \hfill & 2.65\hfill \\ \hfill \sqrt{8}& \approx \hfill & 2.83\hfill \\ \hfill \sqrt{9}& =\hfill & 3\hfill \end{array}

Round \sqrt{17} to two decimal places.

Solution

\begin{array}{cccc}& & & \sqrt{17}\hfill \\ \text{Use the calculator square root key.}\hfill & & & 4.123105626...\hfill \\ \text{Round to two decimal places.}\hfill & & & 4.12\hfill \\ & & & \sqrt{17}\approx 4.12\hfill \end{array}

Round \sqrt{11} to two decimal places.

\approx 3.32

Round \sqrt{13} to two decimal places.

\approx 3.61

Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider \sqrt{9{x}^{2}}. Can you think of an expression whose square is 9{x}^{2}?

\begin{array}{cccccc}\hfill {\left(?\right)}^{2}& =\hfill & 9{x}^{2}\hfill & & & \\ \hfill {\left(3x\right)}^{2}& =\hfill & 9{x}^{2},\hfill & & & \text{so}\phantom{\rule{0.2em}{0ex}}\sqrt{9{x}^{2}}=3x\hfill \end{array}

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

{\left({a}^{m}\right)}^{n}={a}^{m·n}

If we square {a}^{m}, the exponent will become 2m.

{\left({a}^{m}\right)}^{2}={a}^{2m}

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

\begin{array}{cc}\hfill \sqrt{25{u}^{8}}=5{u}^{4}& \text{because}\phantom{\rule{0.2em}{0ex}}{\left(5{u}^{4}\right)}^{2}=25{u}^{8}\hfill \\ \hfill \sqrt{16{r}^{20}}=4{r}^{10}& \text{because}\phantom{\rule{0.2em}{0ex}}{\left(4{r}^{10}\right)}^{2}=16{r}^{20}\hfill \\ \hfill \sqrt{196{q}^{36}}=14{q}^{18}& \text{because}\phantom{\rule{0.2em}{0ex}}{\left(14{q}^{18}\right)}^{2}=196{q}^{36}\hfill \end{array}

Simplify: \sqrt{{x}^{6}} \sqrt{{y}^{16}}.

Solution


\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\sqrt{{x}^{6}}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{\left({x}^{3}\right)}^{2}={x}^{6}.\hfill & & & \phantom{\rule{4em}{0ex}}{x}^{3}\hfill \end{array}


\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\sqrt{{y}^{16}}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{\left({y}^{8}\right)}^{2}={y}^{16}.\hfill & & & \phantom{\rule{4em}{0ex}}{y}^{8}\hfill \end{array}

Simplify: \sqrt{{y}^{8}} \sqrt{{z}^{12}}.

{y}^{4}{z}^{6}

Simplify: \sqrt{{m}^{4}} \sqrt{{b}^{10}}.

{m}^{2}{b}^{5}

Simplify: \sqrt{16{n}^{2}}.

Solution

\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\sqrt{16{n}^{2}}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{\left(4n\right)}^{2}=16{n}^{2}.\hfill & & & \phantom{\rule{4em}{0ex}}4n\hfill \end{array}

Simplify: \sqrt{64{x}^{2}}.

8x

Simplify: \sqrt{169{y}^{2}}.

13y

Simplify: \text{−}\sqrt{81{c}^{2}}.

Solution

\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\text{−}\sqrt{81{c}^{2}}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{\left(9c\right)}^{2}=81{c}^{2}.\hfill & & & \phantom{\rule{4em}{0ex}}-9c\hfill \end{array}

Simplify: \text{−}\sqrt{121{y}^{2}}.

-11y

Simplify: \text{−}\sqrt{100{p}^{2}}.

-10p

Simplify: \sqrt{36{x}^{2}{y}^{2}}.

Solution

\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\sqrt{36{x}^{2}{y}^{2}}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{\left(6xy\right)}^{2}=36{x}^{2}{y}^{2}.\hfill & & & \phantom{\rule{4em}{0ex}}6xy\hfill \end{array}

Simplify: \sqrt{100{a}^{2}{b}^{2}}.

10ab

Simplify: \sqrt{225{m}^{2}{n}^{2}}.

15mn

Simplify: \sqrt{64{p}^{64}}.

Solution

\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\sqrt{64{p}^{64}}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{\left(8{p}^{32}\right)}^{2}=64{p}^{64}.\hfill & & & \phantom{\rule{4em}{0ex}}8{p}^{32}\hfill \end{array}

Simplify: \sqrt{49{x}^{30}}.

7{x}^{15}

Simplify: \sqrt{81{w}^{36}}.

9{w}^{18}

Simplify: \sqrt{121{a}^{6}{b}^{8}}

Solution

\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\sqrt{121{a}^{6}{b}^{8}}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{\left(11{a}^{3}{b}^{4}\right)}^{2}=121{a}^{6}{b}^{8}.\hfill & & & \phantom{\rule{4em}{0ex}}11{a}^{3}{b}^{4}\hfill \end{array}

Simplify: \sqrt{169{x}^{10}{y}^{14}}.

13{x}^{5}{y}^{7}

Simplify: \sqrt{144{p}^{12}{q}^{20}}.

12{p}^{6}{q}^{10}

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\ge 0 to make sure we get the principal square root.

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

\sqrt{36}

6

\sqrt{4}

\sqrt{64}

8

\sqrt{169}

\sqrt{9}

3

\sqrt{16}

\sqrt{100}

10

\sqrt{144}

\text{−}\sqrt{4}

-2

\text{−}\sqrt{100}

\text{−}\sqrt{1}

-1

\text{−}\sqrt{121}

\sqrt{-121}

not a real number

\sqrt{-36}

\sqrt{-9}

not a real number

\sqrt{-49}

\sqrt{9+16}

5

\sqrt{25+144}

\sqrt{9}+\sqrt{16}

7

\sqrt{25}+\sqrt{144}

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

\sqrt{70}

8<\sqrt{70}<9

\sqrt{55}

\sqrt{200}

14<\sqrt{200}<15

\sqrt{172}

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

\sqrt{19}

4.36

\sqrt{21}

\sqrt{53}

7.28

\sqrt{47}

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

\sqrt{{y}^{2}}

y

\sqrt{{b}^{2}}

\sqrt{{a}^{14}}

{a}^{7}

\sqrt{{w}^{24}}

\sqrt{49{x}^{2}}

7x

\sqrt{100{y}^{2}}

\sqrt{121{m}^{20}}

11{m}^{10}

\sqrt{25{h}^{44}}

\sqrt{81{x}^{36}}

9{x}^{18}

\sqrt{144{z}^{84}}

\text{−}\sqrt{81{x}^{18}}

-9{x}^{9}

\text{−}\sqrt{100{m}^{32}}

\text{−}\sqrt{64{a}^{2}}

-8a

\text{−}\sqrt{25{x}^{2}}

\sqrt{144{x}^{2}{y}^{2}}

12xy

\sqrt{196{a}^{2}{b}^{2}}

\sqrt{169{w}^{8}{y}^{10}}

13{w}^{4}{y}^{5}

\sqrt{81{p}^{24}{q}^{6}}

\sqrt{9{c}^{8}{d}^{12}}

3{c}^{4}{d}^{6}

\sqrt{36{r}^{6}{s}^{20}}

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

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?

Writing Exercises

Why is there no real number equal to \sqrt{-64}?

Answers will vary.

What is the difference between {9}^{2} and \sqrt{9}?

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?

Glossary

square of a number

  • If {n}^{2}=m, then m is the square of n
square root of a number

  • If {n}^{2}=m, then n is a square root of m
square root notation

  • If m={n}^{2}, then \sqrt{m}=n. We read \sqrt{m} as ‘the square root of m.’

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Elementary Algebra by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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