Find complex number $K=\frac{(1 + i\sqrt{3})^8}{2^7 (-1 +i\sqrt{3})}$

Question

Can anybody help me with the answer of this question?
$$K=\frac{(1 + i\sqrt{3})^8}{2^7 (-1 +i\sqrt{3})}$$
Hint:
$e^\frac{\pi i}{2}= e^\frac{-3\pi i}{2}=i$
$e^{\pi i}= e^{-\pi i}=-1$
$e^\frac{-\pi i}{2}= e^\frac{3\pi i}{2}=-i$
$e^{2\pi i}= e^{-2\pi i}=1$
How can I convert Cartesian to Polar?

Answer 1

Hint :

$(1 + \sqrt{3}i)^8 = 2^8 \left(\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)^8 = 2^8 \exp(\frac{\pi i}{3})$.

Make the same with denominator and obtain the result.

Answer 2

If $z = 1+i\sqrt{3}$ then $-1+i\sqrt{3} =-\overline {z}$ so you have $$K = {z^8\over -2^7\overline {z}}$$

Notice that $z\overline {z} = 4$ and $z^3= -8$ so $K= {z^9\over -2^9}=...$