Either $a^{2}\neq b^{2}$ or $a^3\neq b^3.$

Hint: Assume that both $a^2=b^2$ and $a^3=b^3$. Using this, can you show that $a=b$?

If so, then for distinct elements $a$ and $b$, it is impossible to satisfy both equalities.

Assume that $a^{3}=b^{3}$ and $a^{2}=b^{2}$. Then $a^{-2}=b^{-2}$. Thus $a^{3}a^{-2}=b^{3}b^{-2}\Rightarrow a=b$.