Answer
$x= 129$
Work Step by Step
The solution is straightforward:
\begin{equation}
\begin{aligned}
(x-4)^{\frac{2}{3}}&=25 \\
\left[ (x-4)^{\frac{2}{3}} \right]^{\frac{3}{2}}&= 25^{\frac{3}{2}}\\
x-4&= \sqrt{25^3}\\
x&= 4+\sqrt{25^3}\\
x&= 4+125\\
x&= 129\\
\end{aligned}
\end{equation}
Check:
\begin{equation}
\begin{aligned}
(129-4)^{\frac{2}{3}}&\stackrel{?}{=}25\\
(125)^{\frac{2}{3}}&\stackrel{?}{=}25\\
\sqrt[3]{125^2}&\stackrel{?}{=}25\\
25&= 25\checkmark\\
\end{aligned}
\end{equation}
The solution is $x= 129$.
