- #1
Santilopez10
- 81
- 8
Homework Statement
Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$
Homework Equations
$$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$
The Attempt at a Solution
First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi k)} = 2^{n-1} 2 e^{i (\frac{2 \pi}{3} +2 \pi k)} $$
this simplifies to $$ e^{i n (\frac {\pi}{3}+ 2\pi k)} = e^{i (\frac{2 \pi}{3} +2 \pi k)} $$
I know how to find an expression for n, but not that it`s only in the field of ## Z ##, any help would be appreciated, thanks!