Answer
$3\sqrt{6}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
5\sqrt{54}-2\sqrt{24}-2\sqrt{96}
,$ 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}
5\sqrt{9\cdot6}-2\sqrt{4\cdot6}-2\sqrt{16\cdot6}
\\\\=
5\sqrt{(3)^2\cdot6}-2\sqrt{(2)^2\cdot6}-2\sqrt{(4)^2\cdot6}
.\end{array}
Extracting the roots of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
5(3)\sqrt{6}-2(2)\sqrt{6}-2(4)\sqrt{6}
\\\\=
15\sqrt{6}-4\sqrt{6}-8\sqrt{6}
.\end{array}
By combining like radicals, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(15-4-8)\sqrt{6}
\\\\=
3\sqrt{6}
.\end{array}
