Answer
The power of the given complex number in the rectangular form is $-4+4\sqrt{3}i$.
Work Step by Step
Consider the given complex number in the polar form,
${{\left[ 2\left( \cos 40{}^\circ +i\sin 40{}^\circ \right) \right]}^{3}}$
If n is a positive integer, then ${{z}^{n}}$ is,
$\begin{align}
& {{z}^{n}}={{\left[ r\left( \cos \theta +i\sin \theta \right) \right]}^{n}} \\
& ={{r}^{n}}\left( \cos n\theta +i\sin n\theta \right)
\end{align}$
That is.,
${{z}^{n}}={{r}^{n}}\left( \cos n\theta +i\sin n\theta \right)$ …… (1)
Now, for the given complex number, using (1) to get,
$\begin{align}
& z={{\left[ 2\left( \cos 40{}^\circ +i\sin 40{}^\circ \right) \right]}^{3}} \\
& z={{2}^{3}}\left( \cos 3\times 40{}^\circ +i\sin 3\times 40{}^\circ \right) \\
& z=8\left( \cos 120{}^\circ +i\sin 120{}^\circ \right) \\
& z=8\left( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right) \\
\end{align}$
Simplifying it further, to get,
$z=-4+4\sqrt{3}i$
Therefore,
The power of the complex number in the rectangular form is $-4+4\sqrt{3}i$.
