calculating complex numbers $ (1+i\sqrt{3})^{2017} + (1-i\sqrt{3})^{2017} $

Question

I have the following $ (1+i\sqrt{3})^{2017} + (1-i\sqrt{3})^{2017} $ and I have to find the answser.

I think the answer might be $ 2^{2017} $ but I don't know how to find

Answer 1

Use $$(1+\sqrt3i)^3=(1-\sqrt3i)^3=-8$$ and $2017\equiv1(\mod3).$

I got $$2\cdot(-8)^{672}=2^{2017}.$$

Answer 2

First, convert these to polar form, so $1+i\sqrt{3}=2e^{i\frac{\pi}{3}},$ and $1-i\sqrt{3}=2e^{-i\frac{\pi}{3}},$ then raise these exponential terms to the desired power, and proceed with the computation, using the identity $$e^{it}=\cos(t)+i\sin(t).$$