Simplifying Expressions With Square Roots
Learning Outcomes
- Simplify expressions with square roots using the order of operations
- Simplify expressions with square roots that contain variables
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: ⓐ [latex]\sqrt{25}+\sqrt{144}[/latex] ⓑ [latex]\sqrt{25+144}[/latex]. Solution| ⓐ Use the order of operations. | |
| [latex]\sqrt{25}+\sqrt{144}[/latex] | |
| Simplify each radical. | [latex]5+12[/latex] |
| Add. | [latex]17[/latex] |
| ⓑ 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
[ohm_question]146635[/ohm_question] [ohm_question]146636[/ohm_question]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-display]\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-display] 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]Answer: Solution Think about what we would have to square to get [latex]{x}^{2}[/latex] . Algebraically, [latex]{\left(?\right)}^{2}={x}^{2}[/latex]
| [latex]\sqrt{{x}^{2}}[/latex] | |
| Since [latex]{\left(x\right)}^{2}={x}^{2}[/latex] | [latex]x[/latex] |
try it
[ohm_question]146637[/ohm_question]example
Simplify: [latex]\sqrt{16{x}^{2}}[/latex].Answer: Solution
| [latex]\sqrt{16{x}^{2}}[/latex] | |
| [latex]\text{Since}{\left(4x\right)}^{2}=16{x}^{2}[/latex] | [latex]4x[/latex] |
try it
[ohm_question]146638[/ohm_question]example
Simplify: [latex]-\sqrt{81{y}^{2}}[/latex].Answer: Solution
| [latex]-\sqrt{81{y}^{2}}[/latex] | |
| [latex]\text{Since}{\left(9y\right)}^{2}=81{y}^{2}[/latex] | [latex]-9y[/latex] |
try it
[ohm_question]146639[/ohm_question]example
Simplify: [latex]\sqrt{36{x}^{2}{y}^{2}}[/latex].Answer: Solution
| [latex]\sqrt{36{x}^{2}{y}^{2}}[/latex] | |
| [latex]\text{Since}{\left(6xy\right)}^{2}=36{x}^{2}{y}^{2}[/latex] | [latex]6xy[/latex] |
try it
[ohm_question]146640[/ohm_question]Licenses & Attributions
CC licensed content, Original
- Question ID 146640, 146639, 146638, 146637. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
