Mathematics » Introducing Decimals » Simplify and Use Square Roots

# Simplifying Variable Expressions with Square Roots

## Simplifying 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.

## Example

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

### Solution

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

 $$\sqrt{{x}^{2}}$$ Since $${\left(x\right)}^{2}={x}^{2}$$ $$x$$

## Example

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

### Solution

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

## Example

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

### Solution

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

## Example

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

### Solution

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

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