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

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

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

Example 6: Adding Square Roots

Add $5\sqrt{12}+2\sqrt{3}\\$ .

Solution

We can rewrite $5\sqrt{12}$ as $5\sqrt{4\cdot 3}$ . According the product rule, this becomes $5\sqrt{4}\sqrt{3}$ . The square root of $\sqrt{4}$ is 2, so the expression becomes $5\left(2\right)\sqrt{3}$ , which is $10\sqrt{3}$ . Now we can the terms have the same radicand so we can add.

$10\sqrt{3}+2\sqrt{3}=12\sqrt{3}$

Try It 6

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

Solution

Example 7: Subtracting Square Roots

Subtract $20\sqrt{72{a}^{3}{b}^{4}c}-14\sqrt{8{a}^{3}{b}^{4}c}$ .

Solution

Rewrite each term so they have equal radicands.

$\begin{array}{ccc}\hfill 20\sqrt{72{a}^{3}{b}^{4}c}& =& 20\sqrt{9}\sqrt{4}\sqrt{2}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 20\left(3\right)\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 120|a|{b}^{2}\sqrt{2ac}\hfill \end{array}$

$\begin{array}{ccc}\hfill 14\sqrt{8{a}^{3}{b}^{4}c}& =& 14\sqrt{2}\sqrt{4}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 14\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 28|a|{b}^{2}\sqrt{2ac}\hfill \end{array}$

Now the terms have the same radicand so we can subtract.

$120|a|{b}^{2}\sqrt{2ac}-28|a|{b}^{2}\sqrt{2ac}\text{= }92|a|{b}^{2}\sqrt{2ac}\text{ }$

Try It 7

Subtract $3\sqrt{80x}-4\sqrt{45x}$ .

Solution

Radicals and Rational Expressions