Question

If $z=\left(\frac{1+i \sqrt{3}}{1+i}\right)^{25}$ and $\arg (z)=\frac{\pi}{k}$ then find $k$

Answer 1

$z =\frac{(1+i \sqrt{3})^{25}}{(1+i)^{25}}=\frac{2^{25} \cdot\left(e^{i \pi / 3}\right)^{25}}{2^{\frac{25}{2}} \cdot\left(e^{i \pi / 4}\right)^{25}}$
$=2^{\frac{25}{2}} \cdot e^{i\left(\frac{25 \pi}{12}\right)}=e^{\frac{25}{2}} \cdot e^{i\left(\frac{\pi}{12}\right)} $
or $\left(2 \pi+\frac{\pi}{12}\right) =\frac{25 \pi}{12}$