Question

Prove that `sqrt(p) + sqrt(q)` is irrational, where p, q are primes.

Answer 1

Let us suppose that `sqrtp + sqrtq` is rational.

Again, let `sqrtp + sqrtq` = a, where a is rational.

Therefore, `sqrtq = a - sqrtp`

On squaring both sides, we get

q = `a^2 + p - 2asqrtp`   .....[∵ (a – b)² = a² + b² – 2ab]

Therefore, `sqrtp = (a^2 + p - q)/(2a)`, which is a contradiction as the right-hand side is rational number while `sqrtp` is irrational, since p is a prime number.

Hence, `sqrtp + sqrtq` is irrational.