Answer
$-11m\sqrt{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
3\sqrt{72m^2}-5\sqrt{32m^2}-3\sqrt{18m^2}
,$ 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}
3\sqrt{36m^2\cdot2}-5\sqrt{16m^2\cdot2}-3\sqrt{9m^2\cdot2}
\\\\=
3\sqrt{(6m)^2\cdot2}-5\sqrt{(4m)^2\cdot2}-3\sqrt{(3m)^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}
3(6m)\sqrt{2}-5(4m)\sqrt{2}-3(3m)\sqrt{2}
\\\\=
18m\sqrt{2}-20m\sqrt{2}-9m\sqrt{2}
.\end{array}
By combining like radicals, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(18m-20m-9m)\sqrt{2}
\\\\=
-11m\sqrt{2}
.\end{array}
