Simplifying Square Roots with Variables
Learning Outcomes
- Simplify square roots with variables
- Recognize that by definition [latex]\sqrt{x^{2}}[/latex] is always nonnegative
Simplifying Square Roots
Radical expressions will sometimes include variables as well as numbers. Consider the expression [latex] \sqrt{9{{x}^{6}}}[/latex]. Simplifying a radical expression with variables is not as straightforward as the examples we have already shown with integers. Consider the expression [latex] \sqrt{{{x}^{2}}}[/latex]. This looks like it should be equal to x, right? Let’s test some values for x and see what happens. In the chart below, look along each row and determine whether the value of x is the same as the value of [latex] \sqrt{{{x}^{2}}}[/latex]. Where are they equal? Where are they not equal? After doing that for each row, look again and determine whether the value of [latex] \sqrt{{{x}^{2}}}[/latex] is the same as the value of [latex]\left|x\right|[/latex].[latex]x[/latex] | [latex]x^{2}[/latex] | [latex]\sqrt{x^{2}}[/latex] | [latex]\left|x\right|[/latex] |
---|---|---|---|
[latex]−5[/latex] | [latex]25[/latex] | [latex]5[/latex] | [latex]5[/latex] |
[latex]−2[/latex] | [latex]4[/latex] | [latex]2[/latex] | [latex]2[/latex] |
[latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] |
[latex]6[/latex] | [latex]36[/latex] | [latex]6[/latex] | [latex]6[/latex] |
[latex]10[/latex] | [latex]100[/latex] | [latex]10[/latex] | [latex]10[/latex] |
Taking the Square Root of a Radical Expression
When finding the square root of an expression that contains variables raised to a power, consider that [latex]\sqrt{x^{2}}=\left|x\right|[/latex]. Examples: [latex]\sqrt{9x^{2}}=3\left|x\right|[/latex], and [latex]\sqrt{16{{x}^{2}}{{y}^{2}}}=4\left|xy\right|[/latex]Example
Simplify. [latex] \sqrt{9{{x}^{6}}}[/latex]Answer: Factor to find identical pairs. [latex-display] \sqrt{3\cdot 3\cdot {{x}^{3}}\cdot {{x}^{3}}}[/latex-display] Rewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify [latex]x^6[/latex] into a square: [latex]{x^3}^2[/latex] [latex-display] \sqrt{{{3}^{2}}\cdot {{\left( {{x}^{3}} \right)}^{2}}}[/latex-display] Separate into individual radicals. [latex-display] \sqrt{{{3}^{2}}}\cdot \sqrt{{{\left( {{x}^{3}} \right)}^{2}}}[/latex-display] Simplify, using the rule that [latex] \sqrt{{{x}^{2}}}=\left|x\right|[/latex]. [latex-display] 3\left|{{x}^{3}}\right|[/latex-display]
Answer
[latex-display] \sqrt{9{{x}^{6}}}=3\left|{{x}^{3}}\right|[/latex-display]Try It
[ohm_question]189413[/ohm_question]Example
Simplify. [latex] \sqrt{100{{x}^{2}}{{y}^{4}}}[/latex]Answer: Separate factors; look for squared numbers and variables. Factor 100 into [latex]10\cdot10[/latex].
[latex] \sqrt{10\cdot 10\cdot {{x}^{2}}\cdot {{y}^{4}}}[/latex]
Factor [latex]y^{4}[/latex] into [latex]\left(y^{2}\right)^{2}[/latex].[latex] \sqrt{10\cdot 10\cdot {{x}^{2}}\cdot {{({{y}^{2}})}^{2}}}[/latex]
Separate the squared factors into individual radicals.[latex] \sqrt{{{10}^{2}}}\cdot \sqrt{{{x}^{2}}}\cdot \sqrt{{{({{y}^{2}})}^{2}}}[/latex]
Take the square root of each radical . Since you do not know whether x is positive or negative, use [latex]\left|x\right|[/latex] to account for both possibilities, thereby guaranteeing that your answer will be positive.[latex]10\cdot\left|x\right|\cdot{y}^{2}[/latex]
Simplify and multiply.[latex]10\left|x\right|y^{2}[/latex]
Answer
[latex-display] \sqrt{100{{x}^{2}}{{y}^{4}}}=10\left| x \right|{{y}^{2}}[/latex-display]Example
Simplify. [latex] \sqrt{49{{x}^{10}}{{y}^{8}}}[/latex]Answer: Look for squared numbers and variables. Factor 49 into [latex]7\cdot7[/latex], [latex]x^{10}[/latex] into [latex]x^{5}\cdot{x}^{5}[/latex], and [latex]y^{8}[/latex] into [latex]y^{4}\cdot{y}^{4}[/latex]. [latex-display] \sqrt{7\cdot 7\cdot {{x}^{5}}\cdot{{x}^{5}}\cdot{{y}^{4}}\cdot{{y}^{4}}}[/latex-display] Rewrite the pairs as squares. [latex-display] \sqrt{{{7}^{2}}\cdot{{({{x}^{5}})}^{2}}\cdot{{({{y}^{4}})}^{2}}}[/latex-display] Separate the squared factors into individual radicals. [latex-display] \sqrt{7^2}\cdot\sqrt{({x^5})^2}\cdot\sqrt{({y^4})^2}[/latex-display] Take the square root of each radical using the rule that [latex] \sqrt{{{x}^{2}}}=\left|x\right|[/latex]. [latex-display] 7\cdot\left|{{x}^{5}}\right|\cdot{{y}^{4}}[/latex-display] Multiply. [latex-display] 7\left|{{x}^{5}}\right|{{y}^{4}}[/latex-display]
Answer
[latex-display] \sqrt{49{{x}^{10}}{{y}^{8}}}=7\left|{{x}^{5}}\right|{{y}^{4}}[/latex-display]Example
Simplify. [latex] \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[/latex]Answer: Factor to find variables with even exponents. [latex-display] \sqrt{{{a}^{2}}\cdot a\cdot {{b}^{4}}\cdot{b}\cdot{{c}^{2}}}[/latex-display] Rewrite [latex]b^{4}[/latex] as [latex]\left(b^{2}\right)^{2}[/latex]. [latex-display] \sqrt{{{a}^{2}}\cdot a\cdot {{({{b}^{2}})}^{2}}\cdot{ b}\cdot{{c}^{2}}}[/latex-display] Separate the squared factors into individual radicals. [latex-display] \sqrt{{{a}^{2}}}\cdot\sqrt{{{({{b}^{2}})}^{2}}}\cdot\sqrt{{{c}^{2}}}\cdot \sqrt{a\cdot b}[/latex-display] Take the square root of each radical. Remember that [latex] \sqrt{{{a}^{2}}}=\left| a \right|[/latex]. [latex-display] \left| a \right|\cdot {{b}^{2}}\cdot\left|{c}\right|\cdot\sqrt{a\cdot b}[/latex-display] Simplify and multiply. The entire quantity [latex] a{{b}^{2}}c[/latex] can be enclosed in the absolute value sign because [latex]b^2[/latex] will be positive anyway. [latex-display] \left| a{{b}^{2}}c \right|\sqrt{ab}[/latex-display]
Answer
[latex-display] \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\left| a{{b}^{2}}c\right|\sqrt{ab}[/latex-display]Contribute!
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CC licensed content, Original
- Screenshot: radical. Provided by: Lumen Learning License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Simplify Square Roots with Variables. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.