Principal argument of $z^n$

I understand the process of finding the principal argument of a complex number if it has been raised to the power of n, i.e., $z^n$.

If we simply had z = x + iy then our principal value is defined to be in the range of $-\pi$< $\theta$< $\pi$. If we have $z^n$ though our principal value changes to $-n\pi$< n$\theta$< n$\pi$ and typically our value would land in this range.

Take the complex number, $z^5$ = $(1 + i)^5$. Our Arg (z) becomes $-5\pi$< 5$\theta$< 5$\pi$.

Can someone kindly explain what it is I am missing? The only thing I can think of is that we define Arg (z) to be $-\pi$< $\theta$< $\pi$ regardless of what z is raised to and so we do not change it but I need convincing.