Question

if p,q are prime position integer, prove that root p+root q is an irrational number

Answer 1

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. Original assumption must be wrong. 

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