Answer
The power of the complex number in the rectangular form is $\frac{i}{32}$.
Work Step by Step
Here,
$z={{\left[ \frac{1}{2}\left( \cos \frac{\pi }{10}+i\sin \frac{\pi }{10} \right) \right]}^{5}}$
Therefore,
$\begin{align}
& z={{\left[ \frac{1}{2}\left( \cos \frac{\pi }{10}+i\sin \frac{\pi }{10} \right) \right]}^{5}} \\
& z=\frac{1}{{{2}^{5}}}\left( \cos 5\times \frac{\pi }{10}+i\sin 5\times \frac{\pi }{10} \right) \\
& z=\frac{1}{32}\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right) \\
& z=\frac{1}{32}\left( 0+i \right) \\
\end{align}$
Simplify it further, to get,
$z=\frac{i}{32}$
The complex number in the rectangular form is $z=\frac{i}{32}$
