## Intermediate Algebra (12th Edition)

$\dfrac{5\sqrt{2}}{3}$
$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $\dfrac{\sqrt{32}}{3}+\dfrac{2\sqrt{2}}{3}-\dfrac{\sqrt{2}}{\sqrt{9}} ,$ find a factor of the radicand that is a perfect power of the index. Then, extract the root of that factor. Finally, combine the like radicals. $\bf{\text{Solution Details:}}$ Rewriting the radicand with a factor that is a perfect power of the index, the given expression is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt{16\cdot2}}{3}+\dfrac{2\sqrt{2}}{3}-\dfrac{\sqrt{2}}{\sqrt{9}} \\\\= \dfrac{\sqrt{(4)^2\cdot2}}{3}+\dfrac{2\sqrt{2}}{3}-\dfrac{\sqrt{2}}{\sqrt{(3)^2}} .\end{array} Extracting the root of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{4\sqrt{2}}{3}+\dfrac{2\sqrt{2}}{3}-\dfrac{\sqrt{2}}{3} .\end{array} Combining the like radicals results to \begin{array}{l}\require{cancel} \dfrac{4\sqrt{2}+2\sqrt{2}-\sqrt{2}}{3} \\\\= \dfrac{5\sqrt{2}}{3} .\end{array}