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.