Prove that √p+√q is an irrational

Question

By using indirect method

Answer 1

Let sqrt p +sqrt q be a rational number a/b ,b is not equal to 0 and in the lowest form.

sqrt p +sqrt q=a/b     

squaring both sides we get,

p+q+2sqrtpq=a^2/b^2

2sqrtpq=(a^2/b^2)-p-q

LHS is irrational and RHS is rational ,which is not possible.

It contradicts  our assumption.

Therefore sqrt p +sqrt q is irrational.