Answer
$a^{2}(b-c)^{}\sqrt[5]{(b-c)^3}$
Work Step by Step
Using the properties of radicals, the given expression, $
\sqrt[5]{a^3(b-c)^4}\sqrt[5]{a^7(b-c)^4}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[5]{a^3(b-c)^4\cdot a^7(b-c)^4}
\\\\=
\sqrt[5]{a^{3+7}(b-c)^{4+4}}
\\\\=
\sqrt[5]{a^{10}(b-c)^{8}}
\\\\=
\sqrt[5]{a^{10}(b-c)^{5}\cdot(b-c)^3}
\\\\=
\sqrt[5]{\left[ a^{2}(b-c)^{} \right]^5\cdot(b-c)^3}
\\\\=
a^{2}(b-c)^{}\sqrt[5]{(b-c)^3}
\end{array}
* Note that it is assumed that no radicands were formed by raising negative numbers to even powers.