# 9.1 Square root expressions

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: understand the concept of square root, be able to distinguish between the principal and secondary square roots of a number, be able to relate square roots and meaningful expressions and to simplify a square root expression.

## Overview

• Square Roots
• Principal and Secondary Square Roots
• Meaningful Expressions
• Simplifying Square Roots

## Square roots

When we studied exponents in Section [link] , we noted that ${4}^{2}=16$ and ${\left(-4\right)}^{2}=16.$ We can see that 16 is the square of both 4 and $-4$ . Since 16 comes from squaring 4 or $-4$ , 4 and $-4$ are called the square roots of 16. Thus 16 has two square roots, 4 and $-4$ . Notice that these two square roots are opposites of each other.

We can say that

## Square root

The square root of a positive number $x$ is a number such that when it is squared the number $x$ results.

Every positive number has two square roots, one positive square root and one negative square root. Furthermore, the two square roots of a positive number are opposites of each other. The square root of 0 is 0.

## Sample set a

The two square roots of 49 are 7 and −7 since

$\begin{array}{lllll}{7}^{2}=49\hfill & \hfill & \text{and}\hfill & \hfill & {\left(-7\right)}^{2}=49\hfill \end{array}$

The two square roots of $\frac{49}{64}$ are $\frac{7}{8}$ and $\frac{-7}{8}$ since

$\begin{array}{lllll}{\left(\frac{7}{8}\right)}^{2}=\frac{7}{8}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{7}{8}=\frac{49}{64}\hfill & \hfill & \text{and}\hfill & \hfill & {\left(\frac{-7}{8}\right)}^{2}=\frac{-7}{8}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{-7}{8}=\frac{49}{64}\hfill \end{array}$

## Practice set a

Name both square roots of each of the following numbers.

36

6 and −6

25

5 and −5

100

10 and −10

64

8 and −8

1

1 and −1

$\frac{1}{4}$

$\frac{1}{2}\text{\hspace{0.17em}}\text{and\hspace{0.17em}}-\frac{1}{2}$

$\frac{9}{16}$

$\frac{3}{4}\text{\hspace{0.17em}}\text{and\hspace{0.17em}}-\frac{3}{4}$

$0.1$

$0.1$ and $-0.1$

$0.09$

$0.03$ and $-0.03$

## Principal and secondary square roots

There is a notation for distinguishing the positive square root of a number $x$ from the negative square root of $x$ .

## Principal square root: $\sqrt{x}$

If $x$ is a positive real number, then

$\sqrt{x}$ represents the positive square root of $x$ . The positive square root of a number is called the principal square root of the number.

## Secondary square root: $-\sqrt{x}$

$-\sqrt{x}$ represents the negative square root of $x$ . The negative square root of a number is called the secondary square root of the number.

$-\sqrt{x}$ indicates the secondary square root of $x$ .

In the expression $\sqrt{x,}$

$\sqrt{\begin{array}{c}\\ \end{array}}$ is called a radical sign .

$x$ is called the radicand .

$\sqrt{x}$ is called a radical .

The horizontal bar that appears attached to the radical sign, $\sqrt{\begin{array}{c}\\ \end{array}}$ , is a grouping symbol that specifies the radicand.

Because $\sqrt{x}$ and $-\sqrt{x}$ are the two square root of $x$ ,

$\begin{array}{lllll}\left(\sqrt{x}\right)\left(\sqrt{x}\right)=x\hfill & \hfill & \text{and}\hfill & \hfill & \left(-\sqrt{x}\right)\left(-\sqrt{x}\right)=x\hfill \end{array}$

## Sample set b

Write the principal and secondary square roots of each number.

$\begin{array}{ccc}9.& & \begin{array}{l}\text{Principal\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is\hspace{0.17em}}\sqrt{9}=3.\\ \text{Secondary\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is}\text{\hspace{0.17em}}-\sqrt{9}=-3.\end{array}\end{array}$

$\begin{array}{ccc}15.& & \begin{array}{l}\text{Principal\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is\hspace{0.17em}}\sqrt{15}.\\ \text{Secondary\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is}\text{\hspace{0.17em}}-\sqrt{15}.\end{array}\end{array}$

Use a calculator to obtain a decimal approximation for the two square roots of 34. Round to two decimal places.
$\begin{array}{l}\text{On}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{Calculater}\\ \begin{array}{lll}\text{Type}\hfill & \hfill & 34\hfill \\ \text{Press}\hfill & \hfill & \begin{array}{||}\hline \sqrt{x}\\ \hline\end{array}\hfill \\ \text{Display reads:}\hfill & \hfill & 5.8309519\hfill \\ \text{Round to 5.83.}\hfill & \hfill & \hfill \end{array}\end{array}$
Notice that the square root symbol on the calculator is $\sqrt{\begin{array}{c}\\ \end{array}}$ . This means, of course, that a calculator will produce only the positive square root. We must supply the negative square root ourselves.

$\begin{array}{lllll}\sqrt{34}\approx 5.83\hfill & \hfill & \text{and}\hfill & \hfill & -\sqrt{34}\approx -5.83\hfill \end{array}$
Note: The symbol ≈ means "approximately equal to."

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