What are the correct steps when defining for which values of $y$ and $x$ an equality is correct?
For instance, $(xy)^3 = xy^3$
What are the correct steps when defining for which values of $y$ and $x$ an equality is correct?
For instance, $(xy)^3 = xy^3$
By commutativity we have $(xy)^3=xyxyxy=x^3y^3$ so rewrite this as $x^3y^3=xy^3$. Subtract the right hand side to get $x^3y-xy=0$ and factor as $x(x+1)(x-1)y=0$, implying $x\in\{0,\pm1\}$ or that $y=0$. The general protocol in these situations is to move everything to one side so that the equation is of the form $\text{blah}=0$, then factor and use the fact that $ab=0$ if and only if $a$ or $b=0$.
(Technically this last property doesn't hold necessarily if we're not talking about real or complex numbers. Take matrices, for example.)