Answer
$2b\sqrt{2b}$
Work Step by Step
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression, $
\dfrac{\sqrt{56ab^3}}{\sqrt{7a}}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt{56ab^3}}{\sqrt{7a}}
\\\\=
\sqrt{\dfrac{56ab^3}{7a}}
\\\\=
\sqrt{8b^3}
.\end{array}
Extracting the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt{8b^3}
\\\\=
\sqrt{4b^2\cdot2b}
\\\\=
\sqrt{(2b)^2\cdot2b}
\\\\=
2b\sqrt{2b}
.\end{array}
Note that all variables are assumed to be positive.