How do I know when to factor out a variable?

[billllib]

In post #9 ,the first step, what algebra rule is that?

 

[haruspex]

In post #9 ,the first step, what algebra rule is that?

You mean this one?
$\gamma=\frac 1{\sqrt{1-\frac{v^2}{c^2}}}$
$\frac 1{\gamma}=\sqrt{1-\frac{v^2}{c^2}}$
The rule there is "do the same to both sides". In this case, the thing I did to both sides was to invert, i.e. turned a fraction like $\frac xy$ into $\frac yx$.