# 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.

*a*can't be negative.

### 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.*

*rationalize the denominator.*

*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]\dfrac{\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]\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]

[embed]## Licenses & Attributions

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- Revision and Adaptation.
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### CC licensed content, Shared previously

- 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. - Question ID 2049.
**Authored by:**Lawrence Morales.**License:**CC BY: Attribution.**License terms:**IMathAS Community License, CC-BY + GPL. - Question ID 110419.
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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.