Answer
$x=\left\{ 8,64 \right\}$
Work Step by Step
Let $z=x^{1/3}$. Then the given expression, $
x^{2/3}-6x^{1/3}=-8
,$ is equivalent to
\begin{array}{l}\require{cancel}
z^2-6z=-8
\\\\
z^2-6z+8=0
\\\\
(z-4)(z-2)=0
\\\\
z=\{2,4\}
.\end{array}
Since $z=x^{1/3}$, then if $z=2$,
\begin{array}{l}\require{cancel}
2=x^{\frac{1}{3}}
\\\\
(2)^3=(x^{\frac{1}{3}})^3
\\\\
8=x
.\end{array}
Since $z=x^{1/3}$, then if $z=4$,
\begin{array}{l}\require{cancel}
4=x^{\frac{1}{3}}
\\\\
(4)^3=(x^{\frac{1}{3}})^3
\\\\
64=x
.\end{array}
Hence, $
x=\left\{ 8,64 \right\}
.$
