#### Answer

$\sqrt[10]{ab^{9}}$

#### Work Step by Step

Using the same indices for the radicals, the given expression, $
\dfrac{\sqrt[]{ab^3}}{\sqrt[5]{a^2b^3}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[2(5)]{a^{1(5)}b^{3(5)}}}{\sqrt[5(2)]{a^{2(2)}b^{3(2)}}}
\\\\=
\dfrac{\sqrt[10]{a^{5}b^{15}}}{\sqrt[10]{a^{4}b^{6}}}
\\\\=
\sqrt[10]{\dfrac{a^{5}b^{15}}{a^{4}b^{6}}}
\\\\=
\sqrt[10]{a^{5-4}b^{15-6}}
\\\\=
\sqrt[10]{a^{1}b^{9}}
\\\\=
\sqrt[10]{ab^{9}}
\end{array}
* Note that it is assumed that all variables represent positive numbers.