Question

prove that root P + root Q is irrational, where p ,q are primes ​

Answer 1

Step-by-step explanation:

First, we'll assume that √p + √q is rational, where p and q are distinct primes

√p + √q = x, where x is rational

Rational numbers are closed under multiplication, so if we square both sides, we still get rational numbers on both sides.

(√p + √q)² = x²

p + 2√(pq) + q = x²

2√(pq) = x² - p - q

√(pq) = (x² - p - q) / 2

Now x, x², p, q and 2 are all rational, and rational numbers are closed under subtraction and division. So (x² - p - q) / 2 is rational.

But since p and q are both primes, then pq is not a perfect square and therefore √(pq) is not rational. But this is a contradiction. The original assumption must be wrong.

So √p + √q is irrational, where p and q are distinct prime