Answer
The power of the complex number in the rectangular form is $z=\frac{27}{2}-i\frac{27\sqrt{3}}{2}$.
Work Step by Step
Here,
$z={{\left[ \sqrt{3}\left( \cos \frac{5\pi }{18}+i\sin \frac{5\pi }{18} \right) \right]}^{6}}$
Therefore,
$\begin{align}
& z={{\left[ \sqrt{3}\left( \cos \frac{5\pi }{18}+i\sin \frac{5\pi }{18} \right) \right]}^{6}} \\
& z={{\left( \sqrt{3} \right)}^{6}}\left( \cos 6\times \frac{5\pi }{18}+i\sin 6\times \frac{5\pi }{6} \right) \\
& z=27\left( \cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3} \right) \\
\end{align}$
Simplify it further to get,
$\begin{align}
& z=27\left( \frac{1}{2}-i\frac{\sqrt{3}}{2} \right) \\
& z=\frac{27}{2}-i\frac{27\sqrt{3}}{2} \\
\end{align}$
The power of the complex number in the rectangular form is $z=\frac{27}{2}-i\frac{27\sqrt{3}}{2}$
