Answer
$\color{blue}{32\sqrt{2} + 32\sqrt{2}i}$
Work Step by Step
$\begin{align*}
& [4(\cos 15^\circ +i\sin 15^\circ)]^3 \\
&= 4^3[\cos (3\cdot 15^\circ) + i\sin (3\cdot 15^\circ)] \quad \text{by De Moivre's Theorem} \\
&= 64[\cos(45^\circ) +i\sin(45^\circ)] \\
&= 64\left( \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) \\
&\color{blue}{= 32\sqrt{2} + 32\sqrt{2}i}
\end{align*}$
