  We investigate the function $d_\mathbf{A}(n)$, which gives the size of a
least size generating set for $\mathbf{A}^n$, in the case where $\mathbf{A}$
has a cube term. We show that if $\mathbf{A}$ has a $k$-cube term and
$\mathbf{A}^k$ is finitely generated, then $d_\mathbf{A}(n) \in O(\log(n))$ if
$\mathbf{A}$ is perfect and $d_\mathbf{A}(n) \in O(n)$ if $\mathbf{A}$ is
imperfect. When $\mathbf{A}$ is finite, then one may replace "Big Oh" with "Big
Theta" in these estimates.
