Let us suppose that √p + √q is rational.
Let √p + √q = a, where a is rational.
=> √q = a – √p
Squaring on both sides, we get
q = a2 + p - 2a√p
=> √p = (a2 + p - q)/2a, which is a contradiction as the right hand side is rational number, while √p is irrational.
Hence, √p + √q is irrational.