Answer
The power of the complex numbers in the rectangular form is $-2-i2\sqrt{3}$.
Work Step by Step
Here,
$z={{\left[ \sqrt{2}\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right) \right]}^{4}}$
Therefore,
$\begin{align}
& z={{\left[ \sqrt{2}\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right) \right]}^{4}} \\
& z=4\left( \cos 4\times \frac{5\pi }{6}+i\sin 4\times \frac{5\pi }{6} \right) \\
& z=4\left( \cos \frac{10\pi }{3}+i\sin \frac{10\pi }{3} \right) \\
\end{align}$
Simplify it further, to get,
$\begin{align}
& z=4\left( -\frac{1}{2}-i\frac{\sqrt{3}}{2} \right) \\
& =-2-i2\sqrt{3}
\end{align}$
The power of the complex number in the rectangular form is $-2-i2\sqrt{3}$
