Question

If p, q are prime positive integers, prove that $\sqrt{p}+\sqrt{q}$ is an irrational number.

Answer 1

Let √p+√q is rational number
A rational number can be written in the form of a/b ( where a and b are co-prime ie they have no common factor other than 1.)
√p+√q=a/b
√p=a/b-√q
√p=(a-b√q)/b

p, q are integers and we supposed that √q is rational then (a-b√q) /b is a rational number.
So, √p is also a rational number.
But this contradicts the fact that √p is an irrational number.
so, our supposition is false
√p+√q is an irrational number