Problem: Let p and q be distinct prime numbers. Suppose $\sqrt{pq}$ is irrational. Prove $\sqrt{p}$ + $\sqrt{q}$ is irrational.
I've done the proof for proving just $\sqrt{p}$ is irrational but I'm just getting so lost here. I've set $\sqrt{p}$ = $\frac{a}{b}$ and $\sqrt{q}$ = $\frac{c}{d}$ then squared both sides so p+q = $\frac{a^2}{b^2}$ + $\frac{c^2}{d^2}$ and I just don't see where I can logicaly go from here.
If $\sqrt{p}+\sqrt{q}$ were rational, then so too would be $$(\sqrt{p}+\sqrt{q})^2=p+q+2\sqrt{pq}$$ Now $p+q+2\sqrt{pq}$ is rational if and only if $\sqrt{pq}$ is, but it isn't.