#### Answer

Choice B

#### Work Step by Step

Multiplying both the numerator and denominator by $
b
,$ the given expression, $
\sqrt[]{\dfrac{a}{b^3}}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[]{\dfrac{a}{b^3}}
\\\\=
\sqrt[]{\dfrac{a}{b^3}\dfrac{b}{b}}
\\\\=
\sqrt[]{\dfrac{a\cdot b}{b^3\cdot b}}
.\end{array}
Hence, $\text{
Choice B
.}$