Question: root 2 plus root 3 is irrational?
Answer: Then its square is rational, because multiplying rationals gives a rational. But (√2+√3)2=2+2√6+3 is irrational because the sum of an irrational and a rational (5) is irrational, so we have a contradiction. You'd have to prove that the sum of two irrational numbers yields an irrational number first.