I recently came across this (somewhat) old MSE question: If $A^3$ and $B^3$ generate a free group, is $\langle A^3,B^3,(AB)^3,(AB^{-1})^3\rangle$ necessarily free?
And this made me wonder—If $A$ and $B$ generate a free group, is $\langle A^3,B^3\rangle=\langle A^3, B^3, (AB)^3, (AB^{-1})^3\rangle$? My intuition tells me no, but I am not able to prove this. Are they the same?