Answer
$2\sqrt{5}$
Work Step by Step
Extracting the factors that are perfect powers of the index, the given expression, $
\sqrt{\sqrt[3]{8000}}
,$ is equivalent to
\begin{align*}\require{cancel}
&
\sqrt{\sqrt[3]{(20)^3}}
\\\\&=
\sqrt{20}
\\\\&=
\sqrt{4\cdot5}
\\\\&=
\sqrt{2^2\cdot5}
\\\\&=
2\sqrt{5}
.\end{align*}
Hence, the given expression simplifies to $
2\sqrt{5}
$.