1. If $\displaystyle r $ is rational ($\displaystyle r \neq 0 $) and $\displaystyle x $ is irrational, prove that $\displaystyle r+x $ and $\displaystyle rx $ are irrational. So $\displaystyle r = \frac{p}{q} $ where $\displaystyle p,q \in \bold{Z} $. So then $\displaystyle \frac{p}{q} + x $ and $\displaystyle \frac{p}{q}x $ somehow have to be irrational.

2. Prove that there is no rational number whose square is $\displaystyle 12 $. So maybe use a proof by contradiction (i.e. assume that $\displaystyle \frac{p^{2}}{q^{2}} = 12 $)?