# 5.7 Simplify and use square roots  (Page 3/8)

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Suppose your calculator has a $\text{10-digit}$ display. Using it to find the square root of $5$ will give $2.236067977.$ This is the approximate square root of $5.$ When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

$\sqrt{5}\approx 2.236067978$

You will seldom use this many digits for applications in algebra. So, if you wanted to round $\sqrt{5}$ to two decimal places, you would write

$\sqrt{5}\approx 2.24$

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

$\begin{array}{ccc}\hfill {2.236067978}^{2}& =& 5.000000002\hfill \\ \hfill {2.24}^{2}& =& 5.0176\hfill \end{array}$

The squares are close, but not exactly equal, to $5.$

Round $\sqrt{17}$ to two decimal places using a calculator.

## Solution

 $\sqrt{17}$ Use the calculator square root key. $4.123105626$ Round to two decimal places. $4.12$ $\sqrt{17}\approx 4.12$

Round $\sqrt{11}$ to two decimal places.

≈ 3.32

Round $\sqrt{13}$ to two decimal places.

≈ 3.61

## Simplify variable expressions with square roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Consider $\sqrt{9{x}^{2}},$ where $x\ge 0.$ Can you think of an expression whose square is $9{x}^{2}?$

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

When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

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

## Solution

Think about what we would have to square to get ${x}^{2}$ . Algebraically, ${\left(?\right)}^{2}={x}^{2}$

 $\sqrt{{x}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(x\right)}^{2}$ $x$

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

y

Simplify: $\sqrt{{m}^{2}}.$

m

Simplify: $\sqrt{16{x}^{2}}.$

## Solution

 $\sqrt{16{x}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(4x\right)}^{2}=16{x}^{2}$ $4x$

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

8 x

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

13 y

Simplify: $-\sqrt{81{y}^{2}}.$

## Solution

 $-\sqrt{81{y}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(9y\right)}^{2}=81{y}^{2}$ $-9y$

Simplify: $-\sqrt{121{y}^{2}}.$

−11 y

Simplify: $-\sqrt{100{p}^{2}}.$

−10 p

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

## Solution

 $\sqrt{36{x}^{2}{y}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(6xy\right)}^{2}=36{x}^{2}{y}^{2}$ $6xy$

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

10 ab

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

15 mn

## Use square roots in applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

## Use a strategy for applications with square roots.

1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Write a complete sentence that answers the question.

## Square roots and area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.

If the area of the square is $A$ square units, the length of a side is $\sqrt{A}$ units. See [link] .

Area (square units) Length of side (units)
$9$ $\sqrt{9}=3$
$144$ $\sqrt{144}=12$
$A$ $\sqrt{A}$

Mike and Lychelle want to make a square patio. They have enough concrete for an area of $200$ square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

## Solution

We know the area of the square is $200$ square feet and want to find the length of the side. If the area of the square is $A$ square units, the length of a side is $\sqrt{A}$ units.

 What are you asked to find? The length of each side of a square patio Write a phrase. The length of a side Translate to an expression. $\sqrt{A}$ Evaluate $\sqrt{A}$ when $A=200$ . $\sqrt{200}$ Use your calculator. $14.142135...$ Round to one decimal place. $\text{14.1 feet}$ Write a sentence. Each side of the patio should be $14.1$ feet.

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