  For $\alpha>-1$, let $A_\alpha^2(\mathbb{B}_N)$ be the weighted Bergman space
on the unit ball $\mathbb{B}_N$ in $\mathbb{C}^N$. In this paper, we prove that
the multiplication operator $M_{z^n}$ is quasi-similar to
$\oplus_1^{\prod_{i=1}^N n_i}M_z$ on $A_\alpha^2(\mathbb{B}_N)$ for the
multi-index $n=(n_1,n_2,\cdots,n_N)$.
