Answer
$\dfrac{8a^{\frac{1}{3}}c^{\frac{2}{3}}}{b^{\frac{5}{12}}}$
Work Step by Step
The given expression, $
\left( \dfrac{64c^{4/3}}{a^{-2/3}b^{5/6}} \right)^{1/2}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{64^{\frac{1}{2}}c^{\frac{4}{3}\cdot\frac{1}{2}}}{a^{\frac{-2}{3}\cdot\frac{1}{2}}b^{\frac{5}{6}\cdot\frac{1}{2}}}
\\\\=
\dfrac{\sqrt{64}c^{\frac{4}{6}}}{a^{\frac{-2}{6}}b^{\frac{5}{12}}}
\\\\=
\dfrac{8c^{\frac{2}{3}}}{a^{\frac{-1}{3}}b^{\frac{5}{12}}}
\\\\=
\dfrac{8a^{\frac{1}{3}}c^{\frac{2}{3}}}{b^{\frac{5}{12}}}
.\end{array}
