We look for perfect cubes that divide evenly into \(108\text{.}\) The easiest way to do this is to try the perfect cubes in order:
\begin{equation*}
1, ~8, ~27, ~64, ~125, \ldots
\end{equation*}
and so on, until we find one that is a factor. For this example, we find that \(108 = 27 \cdot 4\text{.}\) Applying the Product Rule, we write
\begin{align*}
\sqrt[3]{108} \amp =\sqrt[3]{27}\sqrt[3]{4} \amp \amp \blert{\text{Simplify:}~\sqrt[3]{27}=3.}\\
\amp = 3 \sqrt[3]{4}
\end{align*}
This expression is considered simpler than the original radical because the new radicand, \(4\text{,}\) is smaller than the original, \(108\text{.}\)