# 5.7 Simplify and use square roots

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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
• Use square roots in applications

Before you get started, take this readiness quiz.

1. Simplify: ${\left(-9\right)}^{2}.$
If you missed this problem, review Multiply and Divide Integers .
2. Round $3.846$ to the nearest hundredth.
If you missed this problem, review Decimals .
3. Evaluate $12d$ for $d=80.$
If you missed this problem, review Evaluate, Simplify and Translate Expressions .

## Simplify expressions with square roots

To start this section, we need to review some important vocabulary and notation.

Remember that when a number $n$ is multiplied by itself, we can write this as ${n}^{2},$ which we read aloud as $\text{“}\mathit{\text{n}}\phantom{\rule{0.2em}{0ex}}\text{squared.”}$ For example, ${8}^{2}$ is read as $\text{“8}\phantom{\rule{0.2em}{0ex}}\text{squared.”}$

We call $64$ the square of $8$ because ${8}^{2}=64.$ Similarly, $121$ is the square of $11,$ because ${11}^{2}=121.$

## Square of a number

If ${n}^{2}=m,$ then $m$ is the square of $n.$

## Modeling squares

Do you know why we use the word square ? If we construct a square with three tiles on each side, the total number of tiles would be nine.

This is why we say that the square of three is nine.

${3}^{2}=9$

The number $9$ is called a perfect square because it is the square of a whole number.

Doing the Manipulative Mathematics activity Square Numbers will help you develop a better understanding of perfect square numbers

The chart shows the squares of the counting numbers $1$ through $15.$ You can refer to it to help you identify the perfect squares.

## Perfect squares

A perfect square is the square of a whole number.

What happens when you square a negative number?

$\begin{array}{cc}\hfill {\left(-8\right)}^{2}& =\left(-8\right)\left(-8\right)\\ & =64\hfill \end{array}$

When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.

The chart shows the squares of the negative integers from $-1$ to $-15.$

Did you notice that these squares are the same as the squares of the positive numbers?

## Square roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because ${10}^{2}=100,$ we say $100$ is the square of $10.$ We can also say that $10$ is a square root of $100.$

## Square root of a number

A number whose square is $m$ is called a square root of $m.$

If ${n}^{2}=m,$ then $n$ is a square root of $m.$

Notice ${\left(-10\right)}^{2}=100$ also, so $-10$ is also a square root of $100.$ Therefore, both $10$ and $-10$ are square roots of $100.$

So, every positive number has two square roots: one positive and one negative.

What if we only want the positive square root of a positive number? The radical sign, $\sqrt{\phantom{0}},$ stands for the positive square root. The positive square root is also called the principal square root .

## Square root notation

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

$\text{If}\phantom{\rule{0.2em}{0ex}}m={n}^{2},\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\sqrt{m}=n\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}\text{n}\ge 0.$

We can 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.

The chart shows the square roots of the first $15$ perfect square numbers.

Simplify: $\phantom{\rule{0.2em}{0ex}}\sqrt{25}\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}\sqrt{121}.$

## Solution

$\begin{array}{cccc}\text{(a)}\hfill & & & \\ & & & \sqrt{25}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{5}^{2}=25\hfill & & & 5\hfill \\ \\ \text{(b)}\hfill & & & \\ & & & \sqrt{121}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{11}^{2}=121\hfill & & & 11\hfill \end{array}$

Simplify: $\phantom{\rule{0.2em}{0ex}}\sqrt{36}\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}\sqrt{169}.$

1. 6
2. 13

Simplify: $\phantom{\rule{0.2em}{0ex}}\sqrt{16}\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}\sqrt{196}.$

1. 4
2. 14

Every positive number has two square root s and the radical sign indicates the positive one. We write $\sqrt{100}=10.$ If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $-\sqrt{100}=-10.$

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