# How do I find the nth power of a complex number?

Dec 23, 2014

You could use the complex number in rectangular form ($z = a + b i$) and multiply it ${n}^{t h}$ 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 ${n}^{t h}$ power becomes:

${z}^{n} = {r}^{n} \angle n \cdot \theta$

Let's look at an example:
Suppose you want to evaluate ${z}^{4}$ where $z = 4 + 3 i$
Using this notation you should evaluate: ${\left(4 + 3 i\right)}^{4}$ 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 ${1}^{s t}$ graph you can see that:
$a = r \cdot \cos \left(\theta\right)$
$b = r \cdot \sin \left(\theta\right)$

$z = a + b i = r \cdot \cos \left(\theta\right) + r \cdot \sin \left(\theta\right) \cdot i$
$z = - 530 + 331 i$