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.
