z is a complex number and the argument of z^4 is equal to the argument of z. find two possible values between -180 and 180 degrees of the argument of z.
i am not really sure how to do this
I that that the word equal above is used in the sense of equivalence classes.
That is: $z=2\exp \left( {\frac{{2\pi \bf{i}}}{3}} \right)$ and $w=8\exp \left( {\frac{{8\pi\bf{i}}}{3}} \right)$ are different complex numbers they both have the same argument.
(Sorry, but it is against my religion to use degrees.)
So in this question, suppose that $z=r\exp \left(\theta\pi\bf{i} \right)$ so that $z^4=r^4\exp \left(4\theta\pi\bf{i} \right)$.
Solve: $4\theta=\theta+2\pi,~-\pi<\theta\le\pi,~\&~4\theta=\theta+4\pi$ Change to the Devil's degrees.
For the arguments to be the same, you need $4\theta = \theta+2\pi n$ for some integer value of $n$.
This gives $\theta = \dfrac{2\pi n}{3}$. In the interval $-\pi = -180^\circ \le \theta \le 180^\circ = \pi$
So, $-1 \le n \le 1$ to keep $\theta$ in the appropriate range. This gives you three values. $\theta = -\dfrac{2\pi}{3}, \theta = 0, \theta = \dfrac{2\pi}{3}$