I am saying if is the argument of a non-zero complex number then we can write (and conversely). Now if we raise both sides to the we get but and by de Moivre theorem. Thus, we can write as and so is the argument of , i.e. thus . I said "up to a factor of " because as you know there can be many different argument differening by .
The red word factor is probably not the word I'd use since it implies multiplication of the multiples of 2 pi, whereas you are adding the multiples of 2 pi.
If you think of z^n as representing some complex number (1 + i, say) and then think of an Argand diagram showing 1 + i, you should be able to that adding (or subtracting) multiples of 2 pi 'keeps you' at 1 + i. Capisce?