Prove that √p + √q is irrational, where p and q are primes

Solution:

Let us suppose that √p + √q is rational.

Let √p + √q = a where a is rational number.

⇒ √p = a – √q

On squaring both side we get

⇒ $p = a^2+q-2a\sqrt{q}$

[Using formula $(a-b)^2 = a^2+b^2-2ab$ ]

$\Rightarrow 2a\sqrt{q}=a^2+q-p$

$\Rightarrow \sqrt{q} = \dfrac{a^2+q-p}{2a}$

The above statement is a contradiction as the right hand side is a rational number, where the left hand side i.e. √q

is irrational, since p and q are prime numbers.

So, our assumption is wrong.

Hence, √p + √q is rational.