Question

How do I find the `n`th power of a complex number?

Answer 1

You could use the complex number in rectangular form (`z=a+bi`) and multiply it `n^(th)` times by itself but this is not very practical in particular if `n>2`.
What you can do, instead, is to convert your complex number in POLAR form: `z=r angle theta` where `r` is the modulus and `theta` is the argument.
Graphically:

so that now the `n^(th)` power becomes:

`z^n=r^n angle n*theta`

Let's look at an example:
Suppose you want to evaluate `z^4` where `z=4+3i`
Using this notation you should evaluate: `(4+3i)^4` which is difficult and...well...boring!
But if you change it in polar form you get:

Your number in polar form becomes: `z=5 angle 37°` and:
`z^4=5^4 angle (4*37°)=625 angle 148°`

You can now wonder what is the rectangular form of your result.
We get there using the trigonometric form and do some math.
Looking at your `1^(st)` graph you can see that:
`a=r*cos(theta)`
`b=r*sin(theta)`

your complex number becomes now:
`z=a+bi=r*cos(theta)+r*sin(theta)*i`
That gives you:
`z=-530+331i`

(I rounded a little bit to make it clearer)