Understanding the polar form of complex numbers better

Question

let me give an example of what I'm talking about first: say you want to put $(1+i)^{1000}$ into the form a+bi

solution: $2^{500}(e^{iπ/4})^{1000}=2^{500}e^{i250π}=2^{500}e^{i⋅0}=2^{500}$

I don't understand how the $e$ is canceled out, or more specifically how the exponent $π/4 *1000$ is canceled. I realize it has to do with the fact that 4 divides 1000 with no remainder but not much beyond that. Thank you.

Answer 1

It's because\begin{align}e^{i250\pi}&=e^{125\times2\pi i}\\&=\left(e^{2\pi i}\right)^{125}\\&=1^{125}\\&=1.\end{align}

Answer 2

Because $e^{2i\pi}=1$ and $\dfrac{\frac{1000\pi}{4}}{2\pi}$ is an integer. Or, if you prefer, directly by the definition $e^{ix}=\cos x+i\sin x$.