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Study Guides > College Algebra

Adding and Subtracting Square Roots

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 6: Adding Square Roots

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

Solution

We can rewrite 5125\sqrt{12} as 5435\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 6

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

Example 7: Subtracting Square Roots

Subtract 2072a3b4c148a3b4c20\sqrt{72{a}^{3}{b}^{4}c}-14\sqrt{8{a}^{3}{b}^{4}c}.

Solution

Rewrite each term so they have equal radicands.
2072a3b4c=20942aa2(b2)2c=20(3)(2)ab22ac=120ab22ac\begin{array}{ccc}\hfill 20\sqrt{72{a}^{3}{b}^{4}c}& =& 20\sqrt{9}\sqrt{4}\sqrt{2}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 20\left(3\right)\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 120|a|{b}^{2}\sqrt{2ac}\hfill \end{array}
148a3b4c=1424aa2(b2)2c=14(2)ab22ac=28ab22ac\begin{array}{ccc}\hfill 14\sqrt{8{a}^{3}{b}^{4}c}& =& 14\sqrt{2}\sqrt{4}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 14\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 28|a|{b}^{2}\sqrt{2ac}\hfill \end{array}
Now the terms have the same radicand so we can subtract.
120ab22ac28ab22ac92ab22ac 120|a|{b}^{2}\sqrt{2ac}-28|a|{b}^{2}\sqrt{2ac}\text{= }92|a|{b}^{2}\sqrt{2ac}\text{ }

Try It 7

Subtract 380x445x3\sqrt{80x}-4\sqrt{45x}. Solution

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  • College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface. License: CC BY: Attribution.