Basic Technical Mathematics, 11th Edition

11.4 Addition and Subtraction of Radicals

• Similar Radicals • Adding and Subtracting Radicals

When adding or subtracting algebraic expressions, including those with radicals, we combine similar terms. Thus, radicals must be similar, differing only in numerical coefficients, to be added. This means they must have the same order and same radicand.

In order to add radicals, we first express each radical in its simplest form, rationalize any denominators, and then combine those that are similar. For those that are not similar, we can only indicate the addition.

EXAMPLE 1 Adding similar radicals

1. $\begin{array}{ll}2\sqrt{7}\hspace{0.17em}-\hspace{0.17em}5\sqrt{7}\hspace{0.17em}+\hspace{0.17em}\sqrt{7}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}2\sqrt{7}& \text{all similar radicals}\end{array}$

   This result follows the distributive law, as it should. We can write

   $$2\sqrt{7}\hspace{0.17em}-\hspace{0.17em}5\sqrt{7}\hspace{0.17em}+\hspace{0.17em}\sqrt{7}\hspace{0.17em}=\hspace{0.17em}(2\hspace{0.17em}-\hspace{0.17em}5\hspace{0.17em}+\hspace{0.17em}1)\sqrt{7}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}2\sqrt{7}$$

   We can ...