Answer
The power of the complex number ${{\left( \sqrt{2}-i \right)}^{4}}$ in the rectangular form is $z=-7-5.7i$.
Work Step by Step
Consider the given complex number to write in the polar form,
$z={{\left( \sqrt{2}-i \right)}^{4}}$
For a complex number $z=x+iy$, the polar form is given by,
$z=r\left( \cos \theta +i\sin \theta \right)$
Here, $x=r\cos \theta \ \text{ and }\ y=r\sin \theta $, dividing the value of y by x, to get
$\tan \theta =\frac{y}{x}$
Also, the value of r is called the moduli of the complex number, given by,
$\begin{align}
& r=\sqrt{{{x}^{2}}+{{y}^{2}}} \\
& r=\sqrt{{{\left( \sqrt{2} \right)}^{2}}+{{\left( -1 \right)}^{2}}} \\
& r=\sqrt{2+1} \\
& r=\sqrt{3} \\
\end{align}$
For any complex number $z=x+iy$, the sign of the value of x and y determine in which quadrant the value of $z=x+iy$ would lie,
If the value of x is positive and the value of y is positive, then the angle $\theta $ lies in the first quadrant having the value of $\theta $ as $\theta $.
Also, if the value of x is negative and the value of y is positive, then the angle $\theta $ lies in the second quadrant having the value of $\theta $ as $\pi -\theta $.
Also, if the value of x is negative and the value of y is negative, then the angle $\theta $ lies in the third quadrant having the value of $\theta $ as $\pi +\theta $.
Also, if the value of x is positive and the value of y is negative, then the angle $\theta $ lies in the fourth quadrant having the value of $\theta $ as $2\pi -\theta $.
For the given complex number,
$\begin{align}
& \tan \theta =\frac{y}{x} \\
& \tan \theta =\frac{-1}{\sqrt{2}} \\
& \tan \theta =-\frac{1}{\sqrt{2}} \\
& \tan \theta =-\frac{\sqrt{2}}{2} \\
\end{align}$
Since $\theta $ lies in quadrant IV, so
$\begin{align}
& \theta =360{}^\circ -{{\tan }^{-1}}\left( \frac{\sqrt{2}}{2} \right) \\
& =360{}^\circ -35.3{}^\circ \\
& =324.7{}^\circ
\end{align}$
Now,
$z=\sqrt{3}\left( \cos 324.7{}^\circ +i\sin 324.7{}^\circ \right)$
The polar form of the complex number is $z=\sqrt{3}\left( \cos 324.7{}^\circ +i\sin 324.7{}^\circ \right)$
Consider the given complex number in the polar form,
$z=\sqrt{3}\left( \cos 324.7{}^\circ +i\sin 324.7{}^\circ \right)$
If n is a positive integer, then z to the nth power, ${{z}^{n}}$ is
$\begin{align}
& {{z}^{n}}={{\left[ r\left( \cos \theta +i\sin \theta \right) \right]}^{n}} \\
& ={{r}^{n}}\left( \cos n\theta +i\sin n\theta \right)
\end{align}$
${{z}^{n}}={{r}^{n}}\left( \cos n\theta +i\sin n\theta \right)$
Now, for the given complex number,
$\begin{align}
& z={{\left[ \sqrt{3}\left( \cos 324.7{}^\circ +i\sin 324.7{}^\circ \right) \right]}^{4}} \\
& z={{\left( \sqrt{3} \right)}^{4}}\left( \cos 4\times 324.7{}^\circ +i\sin 4\times 324.7{}^\circ \right) \\
& z=9\left( \cos 1298.8{}^\circ +i\sin 1298.8{}^\circ \right) \\
& z\approx -7-5.7i \\
\end{align}$
The power of the complex numbers in the rectangular form is $z=-7-5.7i$
