#### Answer

$a\sqrt[10]{ab^{7}}$

#### Work Step by Step

Using the same indices for the radicals, the given expression, $
\sqrt[5]{a^3b}\sqrt[]{ab}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[5(2)]{a^{3(2)}b^{1(2)}}\cdot\sqrt[2(5)]{a^{1(5)}b^{1(5)}}
\\\\=
\sqrt[10]{a^{6}b^{2}}\cdot\sqrt[10]{a^{5}b^{5}}
\\\\=
\sqrt[10]{a^{6}b^{2}(a^{5}b^{5})}
\\\\=
\sqrt[10]{a^{6+5}b^{2+5}}
\\\\=
\sqrt[10]{a^{11}b^{7}}
\\\\=
\sqrt[10]{a^{10}\cdot ab^{7}}
\\\\=
\sqrt[10]{(a)^{10}\cdot ab^{7}}
\\\\=
a\sqrt[10]{ab^{7}}
\end{array}
* Note that it is assumed that all variables represent positive numbers.