Prove that root p + root q is irrational

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


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.

Some other question

Question 1:Prove that 2-√3 is irrational, given that root 3 is irrational

Question 2:Prove that root 5 is an irrational number

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