Show that the angle of rotation at a nonzero point $z_0 = r_0 e^{i\theta _0 } \in \mathbb{C}$ under the transformation $w = z^n (n = 1, 2, \cdots )$ is $(n-1)\theta _0$ . Show that the scale factor of the transformation at that point is $nr_0 ^{n-1}.$
I know that multiplying complex numbers causes rotation, and that we can calculate the new coordinates on the circle of radius $r_0$, where the points are separated by $\theta _0$ (change the exponential into $\cos \theta + i \sin \theta$ form so that the arguments add as usual since we are multiplying), but I am not sure how to do this with $z^n$ to get the result we want.