### Learning Outcomes

- Simplify expressions with square roots
- Simplify variable expressions with square roots

### Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

### example

Simplify: (a) [latex]\sqrt{25}+\sqrt{144}[/latex] (b) [latex]\sqrt{25+144}[/latex]

Solution

(a) Use the order of operations. | |

[latex]\sqrt{25}+\sqrt{144}[/latex] | |

Simplify each radical. | [latex]5+12[/latex] |

Add. | [latex]17[/latex] |

(b) Use the order of operations. | |

[latex]\sqrt{25+144}[/latex] | |

Add under the radical sign. | [latex]\sqrt{169}[/latex] |

Simplify. | [latex]13[/latex] |

### try it

Notice the different answers in parts (a) and (b) of the example above. It is important to follow the order of operations correctly. In (a), we took each square root first and then added them. In (b), we added under the radical sign first and then found the square root.

## 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 [latex]\sqrt{9{x}^{2}}[/latex], where [latex]x\ge 0[/latex]. Can you think of an expression whose square is [latex]9{x}^{2}?[/latex]

[latex]\begin{array}{ccc}\hfill {\left(?\right)}^{2}& =& 9{x}^{2}\hfill \\ \hfill {\left(3x\right)}^{2}& =& 9{x}^{2}\text{ so }\sqrt{9{x}^{2}}=3x\hfill \end{array}[/latex]

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: [latex]\sqrt{{x}^{2}}[/latex], where [latex]x\ge 0[/latex]

### try it

### example

Simplify: [latex]\sqrt{16{x}^{2}}[/latex], where [latex]x\ge 0[/latex]

### try it

### example

Simplify: [latex]-\sqrt{81{y}^{2}}[/latex], where [latex]y\ge 0[/latex]

### try it

### example

Simplify: [latex]\sqrt{36{x}^{2}{y}^{2}}[/latex], where [latex]x\ge 0[/latex] and [latex]y\ge 0[/latex]