Answer
$24\sqrt{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
6\sqrt{18}-\sqrt{32}+2\sqrt{50}
,$ 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}
6\sqrt{9\cdot2}-\sqrt{16\cdot2}+2\sqrt{25\cdot2}
\\\\
6\sqrt{(3)^2\cdot2}-\sqrt{(4)^2\cdot2}+2\sqrt{(5)^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}
6(3)\sqrt{2}-4\sqrt{2}+2(5)\sqrt{2}
\\\\=
18\sqrt{2}-4\sqrt{2}+10\sqrt{2}
.\end{array}
By combining like radicals, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(18-4+10)\sqrt{2}
\\\\=
24\sqrt{2}
.\end{array}
