# Prove that for rational nonzero $a, b,$ the expression $a\sqrt3 + b\sqrt5$ is irrational.

I've proven the case where only one of $a$, $b$ is zero. But this is the proof for both $a$ and $b$ nonzero. This is what I have:

Suppose $a\sqrt3 + b\sqrt5 = X$ for $X$ rational. Squaring,

$X^{2}$ = $3a^{2} + 2(ab\sqrt3 \sqrt5$) + $5b^{2}$