Question

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

Answer 1

Answer: 12.00

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