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Study Guides > College Algebra

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 2\sqrt{2} and 323\sqrt{2} is 424\sqrt{2}. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression 18\sqrt{18} can be written with a 22 in the radicand, as 323\sqrt{2}, so 2+18=2+32=42\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}.

How To: Given a radical expression requiring addition or subtraction of square roots, solve.

  1. Simplify each radical expression.
  2. Add or subtract expressions with equal radicands.

Example: Adding Square Roots

Add 512+235\sqrt{12}+2\sqrt{3}.

Answer: We can rewrite 5125\sqrt{12} as 54⋅35\sqrt{4\cdot 3}. According the product rule, this becomes 5435\sqrt{4}\sqrt{3}. The square root of 4\sqrt{4} is 2, so the expression becomes 5(2)35\left(2\right)\sqrt{3}, which is 10310\sqrt{3}. Now we can the terms have the same radicand so we can add.

103+23=12310\sqrt{3}+2\sqrt{3}=12\sqrt{3}

Try It

Add 5+620\sqrt{5}+6\sqrt{20}.

Answer: 13513\sqrt{5}

in the next video we show more examples of how to subtract radicals. https://youtu.be/77TR9HsPZ6M

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 bcb\sqrt{c}, multiply by cc\frac{\sqrt{c}}{\sqrt{c}}. 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 a+bca+b\sqrt{c}, then the conjugate is a−bca-b\sqrt{c}.

How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.

  1. Multiply the numerator and denominator by the radical in the denominator.
  2. Simplify.

Example: Rationalizing a Denominator Containing a Single Term

Write 23310\frac{2\sqrt{3}}{3\sqrt{10}} in simplest form.

Answer: The radical in the denominator is 10\sqrt{10}. So multiply the fraction by 1010\frac{\sqrt{10}}{\sqrt{10}}. Then simplify.

23310⋅1010 23030 3015\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}

Try It

Write 1232\frac{12\sqrt{3}}{\sqrt{2}} in simplest form.

Answer: 666\sqrt{6}

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