Solving “coin exchange” for coins of power values by greedy algorithm

Question

When solving the problem of coin exchange by greedy algorithm, why will we will always have the correct result if the coin values are $1, a, a^2, \cdots, a^n$, where $a\ge 2$ and $n\gt 0$?

For example, if $a = 3$, $n =3$, we get the following coin values: 1, 3, 9, 27. When total exchange is 16, answer is 4 coins, as returned by greedy algorithm, $16=9+3+3+1$.