Let $\tau$ be a linear map on a finite dimensional complex vector space. Clearly, if $\lambda$ is an eigenvalue of $\tau$ then $\lambda^n$ is an eigenvalue of $\tau^n$, for any natural (integer, on condition $\tau$ is invertible) number $n$. It easily follows from Jordan theorem, that every eigenvalue of $\tau^n$ has to be of the form $\lambda^n$.
I have to convince students who have only basic knowledge about linear algebra that the above statement is true.
Is there any elementary explanation of this fact without using Jordan theorem?