How can I prove this equation? My prof put this equation on the board, without any kind of explanation or proof. When I asked him for one, he didn't really give me a solid answer.

$$w = r(\cos \theta + i\sin \theta)$$

Then

$w^n = r^n(\cos n\theta + i\sin n\theta)$ for $n \in \mathbb{R}$

I would try to prove this, but it's a bit beyond me considering I learn't what a complex number is a week ago. Please don't use Euler's Theorem. 

EDIT: I ended up using induction since the formula is not valid for non-integers.
I don't know if you've been taught the form $e^{i\theta} = cos(\theta) + i sin(\theta$).
If yes, then $$w = re^{i\theta}$$ and $$w^n = r^ne^{in\theta} = r^n(cos(n\theta) + i sin(n\theta))$$

Maybe your prof did not give you the explanation because he haven't taught you this form yet. I don't remember very well what comes first in the math program between complex numbers and exponentials (plus it can depend on the country).
I don't know if you've been taught the form $e^{i\theta} = cos(\theta) + i sin(\theta$).
If yes, then $$w = re^{i\theta}$$ and $$w^n = r^ne^{in\theta} = r^n(cos(n\theta) + i sin(n\theta))$$

Maybe your prof did not give you the explanation because he haven't taught you this form yet. I don't remember very well what comes first in the math program between complex numbers and exponentials (plus it can depend on the country).