#### Answer

$\dfrac{\sqrt[5]{48a^4b^3}}{2b^{2}}$

#### Work Step by Step

Rationalizing the denominator of the given expression, $
\dfrac{\sqrt[5]{3a^4}}{\sqrt[5]{2b^7}}
,$ we find:
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[5]{3a^4}}{\sqrt[5]{2b^7}}\cdot\dfrac{\sqrt[5]{16b^3}}{\sqrt[5]{16b^3}}
\\\\=
\dfrac{\sqrt[5]{48a^4b^3}}{\sqrt[5]{32b^{10}}}
\\\\=
\dfrac{\sqrt[5]{48a^4b^3}}{\sqrt[5]{(2b^{2})^5}}
\\\\=
\dfrac{\sqrt[5]{48a^4b^3}}{2b^{2}}
.\end{array}
* Note that it is assumed that all variables represent positive numbers.