Answer
$32\sqrt{2}+i32\sqrt{2}$.
Work Step by Step
Here,
$z={{\left[ 4\left( \cos {{15}^{{}^\circ }}+i\sin {{15}^{{}^\circ }} \right) \right]}^{3}}$
$\begin{align}
& z={{4}^{3}}\left( \cos 3\times {{15}^{{}^\circ }}+i\sin 3\times {{15}^{{}^\circ }} \right) \\
& z=64\left( \cos {{45}^{{}^\circ }}+i\sin {{45}^{{}^\circ }} \right) \\
& z=64\left( \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} \right) \\
\end{align}$
Simplifying it further, to get,
$z=32\sqrt{2}+i32\sqrt{2}$
Therefore the complex number in the rectangular form is $32\sqrt{2}+i32\sqrt{2}$
