By Definition SIM we know that there is a nonsingular matrix $S$ so that $A=\similar{B}{S}$. Then
\begin{align*}
A^3&=(\similar{B}{S})^3\\
&=(\similar{B}{S})(\similar{B}{S})(\similar{B}{S})\\
&=\inverse{S}B(S\inverse{S})B(S\inverse{S})BS&&\knowl{./knowls/theorem.MMA.knowl}{\text{Theorem MMA}}\\
&=\inverse{S}B(I_n)B(I_n)BS&&\knowl{./knowls/definition.MI.knowl}{\text{Definition MI}}\\
&=\inverse{S}BBBS&&\knowl{./knowls/theorem.MMIM.knowl}{\text{Theorem MMIM}}\\
&=\inverse{S}B^3S
\end{align*}
This equation says that $A^3$ is similar to $B^3$ (via the matrix $S$).
More generally, if $A$ is similar to $B$, and $m$ is a non-negative integer, then $A^m$ is similar to $B^m$. This can be proved using induction (Proof Technique I).