Thread: Complex Numbers: How to solve these kind of tasks?

I'm having problems to solve these kind of tasks with powers - like:

$\displaystyle \left ( \frac{\sqrt{3}-i}{1+i} \right )^{24}$

Or

$\displaystyle \frac{(\sqrt{3}-i)^{13}}{(1+i)^{16}} + \frac{(\sqrt{3}+i)^{13}}{(1-i)^{16}}$

I try with racitonalizing but i don't get much. Would appreciate any kind of help.
Thanks,

Flamuri

Originally Posted by Flamuri

$\displaystyle \left ( \frac{\sqrt{3}-i}{1+i} \right )^{24}$

First use polar form.
$\displaystyle \sqrt{3}-i=2\exp\left(\frac{-\pi i}{6}\right)~\&~1+i=\sqrt{2}\exp\left(\frac{\pi i}{4}\right)$

Now it is easy to raise those to 24th power.