Answer
$16\sqrt[4]{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
2\sqrt[4]{512}+4\sqrt[4]{32}
,$ simplify first each term by expressing the radicand as a factor that is a perfect power of the index. Then, extract the root. Finally, combine the like radicals.
$\bf{\text{Solution Details:}}$
Expressing the radicand as an expression that contains a factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
2\sqrt[4]{256\cdot2}+4\sqrt[4]{16\cdot2}
\\\\=
2\sqrt[4]{(4)^2\cdot2}+4\sqrt[4]{(2)^2\cdot2}
.\end{array}
Extracting the roots of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
2(4)\sqrt[4]{2}+4(2)\sqrt[4]{2}
\\\\=
8\sqrt[4]{2}+8\sqrt[4]{2}
.\end{array}
By combining like radicals, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(8+8)\sqrt[4]{2}
\\\\=
16\sqrt[4]{2}
.\end{array}
