Answer
$x=\{ 1,125 \}$
Work Step by Step
Let $z=x^{1/3}$. Then the given expression, $
x^{2/3}-6x^{1/3}+5=0
,$ is equivalent to
\begin{array}{l}\require{cancel}
z^2-6z+5=0
\\\\
(z-5)(z-1)=0
\\\\
z=\{1,5\}
.\end{array}
Since $z=x^{1/3}$, then if $z=1$,
\begin{array}{l}\require{cancel}
1=x^{\frac{1}{3}}
\\\\
(1)^3=(x^{\frac{1}{3}})^3
\\\\
1=x
.\end{array}
Since $z=x^{1/3}$, then if $z=5$,
\begin{array}{l}\require{cancel}
5=x^{\frac{1}{3}}
\\\\
(5)^3=(x^{\frac{1}{3}})^3
\\\\
125=x
.\end{array}
Hence, $
x=\{ 1,125 \}
.$
