Prove that if x is rational and x≠0, then 1/x is rational.
To get a common factor in both numerator and denominator.
It helps to arrive at the final fraction since $\displaystyle \frac{b}{b}=1$ and $\displaystyle \displaystyle\frac{\left(\frac{1}{b}\right)}{\left (\frac{1}{b}\right)}=1$
You could also write
$\displaystyle \displaystyle\frac{1}{\left(\frac{a}{b}\right)}=\f rac{b}{b}\frac{1}{\left(\frac{a}{b}\right)}=\frac{ b}{a}$