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Complex Number Power Formula

A complex number is the form of $x$ + $y^{i}$, where x and y are the real numbers and i is an imaginary number. The i satisfies $i^{2}$ = $-1$.

The complex number power formula is given below.

\[\LARGE z^{n}=(re^{i\theta})^{n}=r^{n}e^{in\theta}\]

Solved Examples
Question 1:Compute: $(3+3i)^{5}$

Solution:

Here is the exponential form of 3+3i

$r=\sqrt{9+9}=3\sqrt{2}$

$\tan\theta=\frac{3}{3}\Rightarrow\arg z=\frac{\pi}{4}$

$3+3i=3\sqrt{2}e^{i\frac{\pi}{4}}$

Now, $(3+3i)^{5}=(3\sqrt{2})^{5}e^{i\frac{5\pi}{4}}$

$=972\sqrt{2}\left ( \cos \left ( \frac{5\pi}{4} \right )+i\sin \left ( \frac{5\pi}{4} \right )\right )$

$=972\sqrt{2}\left ( -\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i\right )$

$=-972-972i$