Q:

Find modulus and argument

The complex numbers $p$ and $q$ are such that :
 $|p| = 2,$  $\arg(p) = \pi/3$  $|q| = 7,$  and $\arg (q) = -\pi/4.$ 
Find the modulus and argument of $(p/q) ^5 $
I found that: $p = 1 + i \sqrt{3}$ and
 $q = (7 \sqrt{2} ) /2) - (7 \sqrt{2} ) /2)i.$

A:

$$|\frac {p}{q}|=\frac {|p|}{|q|} $$
$$=\frac {2}{7} $$
$$|\frac {p}{q}|^5=\frac {2^5}{7^5} $$
$$arg (\frac {p}{q})=arg (p)-arg (q) $$
$$=\frac {\pi}{3}-\frac {-\pi}{4} $$
$$=\frac {7\pi}{12} $$
$$arg ((\frac {p}{q})^5)=5arg (\frac {p}{q}) $$
$$=\frac {35\pi}{12}=\frac {11\pi}{12} $$

