Answer
$x= 1032 $
Work Step by Step
Given\begin{equation}
\sqrt[5]{x-8}=4.
\end{equation} Start by rewriting the radical as a power of fraction and then solve for $x$.
\begin{equation}
\begin{aligned}
\left(x-8\right)^{\frac{1}{5}}&= 4\\
\left(\left(x-8\right)^{\frac{1}{5}}\right)^3& = \left( 4\right)^5 \\
x-8& =1024\\
x&= 1024+8\\
& = 1032.
\end{aligned}
\end{equation}Check.
\begin{equation}
\begin{aligned}
\left( 1032-8\right)^{\frac{1}{5}}&\stackrel{?}{=}4\\
4& =4\quad \checkmark.
\end{aligned}
\end{equation} The solution is $x= 1032 $.
