Operations on Square Roots
Learning Outcomes
- Add and subtract square roots.
- Rationalize denominators.
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]
[embed]Example: Subtracting Square Roots
Subtract [latex]20\sqrt{72{a}^{3}{b}^{4}c}-14\sqrt{8{a}^{3}{b}^{4}c}[/latex].Answer: Rewrite each term so they have equal radicands.
Try It
Subtract [latex]3\sqrt{80x}-4\sqrt{45x}[/latex].Answer: [latex-display]0[/latex-display]
[embed]Rationalize Denominators
Recall the identity property of addition
We leverage an important and useful identity in this section in a technique commonly used in college algebra:rewriting an expression by multiplying it by a well-chosen form of the number 1.
Because the additive identity states that [latex]a\cdot1=a[/latex], we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we rationalize the denominator.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]\dfrac{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]\dfrac{\sqrt{10}}{\sqrt{10}}[/latex]. Then simplify.
Try It
Write [latex]\dfrac{12\sqrt{3}}{\sqrt{2}}[/latex] in simplest form.Answer: [latex-display]6\sqrt{6}[/latex-display]
[embed]How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.
- Find the conjugate of the denominator.
- Multiply the numerator and denominator by the conjugate.
- Use the distributive property.
- Simplify.
Example: Rationalizing a Denominator Containing Two Terms
Write [latex]\dfrac{4}{1+\sqrt{5}}[/latex] in simplest form.Answer: Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\sqrt{5}[/latex] is [latex]1-\sqrt{5}[/latex]. Then multiply the fraction by [latex]\dfrac{1-\sqrt{5}}{1-\sqrt{5}}[/latex].
Try It
Write [latex]\dfrac{7}{2+\sqrt{3}}[/latex] in simplest form.Answer: [latex-display]14 - 7\sqrt{3}[/latex-display]
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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].
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