It is easy to see that for each perfect power $N = n^k$ it holds that
$$N-1 = (n-1)\sum_{0\leq i<k} n^{i}$$
It is not much harder to see that for each $x < N$ there are $a_i < n$ such that
$$x = \sum_{0\leq i<k} a_i\ n^{i}$$
What makes perfect powers perfect bases for the redundancy-free representation of numbers $x<N$ is that there is a perfect 1:1 correspondence between all $x<N$ and all k-tuples $\{a_i\}$ with $a_i<n$.
I wonder why these - as I find important - facts are not more often prominently mentioned, e.g. in the Wikipedia article on perfect powers.