#### Answer

$a^{2}b^{2}c^{2}\sqrt[6]{a^2bc^2}$

#### Work Step by Step

Using the same indices for the radicals, the given expression, $
\sqrt[]{a^4b^3c^4}\sqrt[3]{ab^2c}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[2(3)]{a^{4(3)}b^{3(3)}c^{4(3)}}\cdot\sqrt[3(2)]{a^{1(2)}b^{2(2)}c^{1(2)}}
\\\\=
\sqrt[6]{a^{12}b^{9}c^{12}}\cdot\sqrt[6]{a^{2}b^{4}c^{2}}
\\\\=
\sqrt[6]{a^{12}b^{9}c^{12}(a^{2}b^{4}c^{2})}
\\\\=
\sqrt[6]{a^{12+2}b^{9+4}c^{12+2}}
\\\\=
\sqrt[6]{a^{14}b^{13}c^{14}}
\\\\=
\sqrt[6]{a^{12}b^{12}c^{12}\cdot a^2bc^2}
\\\\=
\sqrt[6]{(a^{2}b^{2}c^{2})^6\cdot a^2bc^2}
\\\\=
a^{2}b^{2}c^{2}\sqrt[6]{a^2bc^2}
\end{array}
* Note that it is assumed that all variables represent positive numbers.