## Simplifying Variable Expressions with Square Roots

Contents

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 \(9{x}^{2}?\)

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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