Let $A$ be an associative finitely-generated algebra. I want to show that any power of $A$ is finitely-generated as well.
Definition of power: $$A^k \stackrel{\text{def}}= \text{ { all finite sums $\sum_{i_1, \dots, _k} a_{i_1} \dots a_{i_k}$ : $a_{i1},\dots, a_{ik} \in A$ } }.$$ in other words: $A^k$ is set of all sums of the form $a_1, \dots, a_k$, where $a_j \in A$, for all $j \in [1; k]$