Answer
$-49p\sqrt{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
9\sqrt{27p^2}-14\sqrt{108p^2}+2\sqrt{48p^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}
9\sqrt{9p^2\cdot3}-14\sqrt{36p^2\cdot3}+2\sqrt{16p^2\cdot3}
\\\\=
9\sqrt{(3p)^2\cdot3}-14\sqrt{(6p)^2\cdot3}+2\sqrt{(4p)^2\cdot3}
.\end{array}
Extracting the roots of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
9(3p)\sqrt{3}-14(6p)\sqrt{3}+2(4p)\sqrt{3}
\\\\=
27p\sqrt{3}-84p\sqrt{3}+8p\sqrt{3}
.\end{array}
By combining like radicals, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(27p-84p+8p)\sqrt{3}
\\\\=
-49p\sqrt{3}
.\end{array}
