Find Jordan form of powers of Jordan matrix

Question

Let $A$ be a Jordan matrix with blocks $J_5(0),J_6(0)$ with $J_m(\lambda)$ having size $m\times m$. I am to find the Jordan form of $A^2$. Since $A$ is in Jordan form and powers of $A$ have the same transition matrices $P$ as $A$, I think this implies $A^2$ must be in Jordan form, and to find it I have no choice but to calculate the squares of $J_5(0),J_6(0)$. Is this correct, and is there an easier way?