Let us suppose that √3 + √5 is rational. Let √3 = √5 − a, were a is rational.

On squaring both sides, we get

(√3)² = (√5 – a )²

⇒ 3 = 5 + a² − 2a√5 [ Using (a-b)² = a² + b² + 2ab]

⇒ 2a√5 = 2 + a²

This is not possible because right hand side is rational while left hand side i.e. √5 is irrational.

So, our assumption is wrong. √3 + √5 is irrational.