Last Digits of a power always the same

Question

Let $a$ be a number $>2$, $b$ the base and $e>1$ some exponent. Then it seems that $a^{i\cdot b^e}$ with $i=1,2,3,...$ in base $b=10$ (the number also represented in base 10) always have the same $e+1$ last digits. It also seems to work in base $b=9$ and $b=8$ (up to the last $e$ digits for some $a$; the $e+1$ digits follows some cycle) but fails for lower bases. What is the reason behind this?