Answer
$\dfrac{10a^2}{3b^3}$
Work Step by Step
Extracting the factors that are perfect powers of the index, the given expression simplifies to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{100a^4}{9b^6}}
\\\\=
\sqrt{\left( \dfrac{10a^2}{3b^3}\right)^2}
\\\\=
\dfrac{10a^2}{3b^3}
.\end{array}
Note that all variables are assumed to represent positive numbers.
