Prove that 
is irrational, where p and q are primes.
Let us suppose that √p + √q is rational. Let √p = √q − a, were a is rational.
On squaring both sides, we get
(√p)2 = (√q – a )2
⇒ p2 = q2 + a2 − 2a√q [ Using (a-b)2 = a2 + b2 + 2ab]
⇒ 2a√q = p2 + q2 + a2
This is not possible because right hand side is rational while left hand side i.e. √q is irrational.
So, our assumption is wrong. √p + √q is irrational.
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