Do there exist positive irrational numbers $x$ and $y$ such that $x+y$ and $x y$ are both rational? If so, give an example; if not, explain why not.
Such numbers do exist. One example is $x=3-\sqrt{2}$ and $y=3+\sqrt{2}$ (which are irrational due to the famous fact that $\sqrt{2}$ is irrational). Then $x+y=6$ and $x y=7$.