Exponential Function with big exponents

Question

Does somebody know how I can simplify something like $e^{2022*i*\pi/4}$ or $e^{2022*i*\pi/8}$ by hand? I tried to image it on the unit circle. Without the 4 or 8 it would be easy since 2022 is a multiple of 2. But what is the intuition if the whole exponent gets divided?

Answer 1

For the above 2 examples you provided let the expression be $e^{ix}$

So, $x=\frac{2022\pi}{4}=504\pi+\frac{3}{2}\pi$.

Therefore, by Euler's Formula

$e^{ix}=\cos x+i\sin x=0-i=-i$

For 2nd case $x=\frac{2022\pi}{8}=252\pi+\frac{3\pi}{4}$

Therefore, by Euler's Formula

$e^{ix}=\cos x+i\sin x=-\frac{1}{\sqrt2}+i\frac{1}{\sqrt2}$