# 9.1 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

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 [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:
$\text{−}\sqrt{100}$ $\sqrt{-100}$ .
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 ${\left(-15\right)}^{2}=225$ 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

$\sqrt{m}$ 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 [link] to make a table of square roots you can refer to as you work this chapter.

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: $\text{−}\sqrt{49}$ $\sqrt{225}$ .

$-7$ $15$

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 [link] below.

how to square
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