Operations on Square Roots
Learning Objectives
- Add and subtract square roots
- Rationalize denominators
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\sqrt{2}[/latex] and [latex]3\sqrt{2}[/latex] is [latex]4\sqrt{2}[/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\sqrt{18}[/latex] can be written with a [latex]2[/latex] in the radicand, as [latex]3\sqrt{2}[/latex], so [latex]\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}[/latex].
How To: Given a radical expression requiring addition or subtraction of square roots, solve.
- Simplify each radical expression.
- Add or subtract expressions with equal radicands.
Example: Adding Square Roots
Add [latex]5\sqrt{12}+2\sqrt{3}[/latex].Answer: We can rewrite [latex]5\sqrt{12}[/latex] as [latex]5\sqrt{4\cdot 3}[/latex]. According the product rule, this becomes [latex]5\sqrt{4}\sqrt{3}[/latex]. The square root of [latex]\sqrt{4}[/latex] is 2, so the expression becomes [latex]5\left(2\right)\sqrt{3}[/latex], which is [latex]10\sqrt{3}[/latex]. Now we can the terms have the same radicand so we can add.
[latex]10\sqrt{3}+2\sqrt{3}=12\sqrt{3}[/latex]
Try It
Add [latex]\sqrt{5}+6\sqrt{20}[/latex].Answer: [latex-display]13\sqrt{5}[/latex-display]
Rationalize Denominators
When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator. We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical. For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex]b\sqrt{c}[/latex], multiply by [latex]\frac{\sqrt{c}}{\sqrt{c}}[/latex]. For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex]a+b\sqrt{c}[/latex], then the conjugate is [latex]a-b\sqrt{c}[/latex].How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.
- Multiply the numerator and denominator by the radical in the denominator.
- Simplify.
Example: Rationalizing a Denominator Containing a Single Term
Write [latex]\frac{2\sqrt{3}}{3\sqrt{10}}[/latex] in simplest form.Answer: The radical in the denominator is [latex]\sqrt{10}[/latex]. So multiply the fraction by [latex]\frac{\sqrt{10}}{\sqrt{10}}[/latex]. Then simplify.
[latex]\begin{array}{l}\frac{2\sqrt{3}}{3\sqrt{10}}\cdot \frac{\sqrt{10}}{\sqrt{10}}\text{ }\\ \frac{2\sqrt{30}}{30}\text{ }\\ \frac{\sqrt{30}}{15}\end{array}[/latex]
Try It
Write [latex]\frac{12\sqrt{3}}{\sqrt{2}}[/latex] in simplest form.Answer: [latex]6\sqrt{6}[/latex]
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- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Adding Radicals Requiring Simplification. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Subtracting Radicals. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
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- Question ID 3441. Authored by: Jessica Reidel. License: CC BY: Attribution. License terms: IMathAS Community License, CC-BY + GPL.
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- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
