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 6: Adding Square Roots
Add [latex]5\sqrt{12}+2\sqrt{3}\\[/latex].
Solution
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]
Example 7: Subtracting Square Roots
Subtract [latex]20\sqrt{72{a}^{3}{b}^{4}c}-14\sqrt{8{a}^{3}{b}^{4}c}[/latex].
Solution
Rewrite each term so they have equal radicands.
Now the terms have the same radicand so we can subtract.