BdMO National Secondary 2011/9

[sakib.creza]

Can it be generally said that $\sqrt{a}+\sqrt{b}$ and $\sqrt{a}-\sqrt{b}$ is always irrational if both $\sqrt{a}$ and $\sqrt{b}$ are irrational? I think it is because in Secondary and higher Secondary Marathon Adib Bhai had asked for a similar proof which Sanzeed Bhai proved. Also, during the olympiad is it necessary to prove the aforementioned statement in order to prove this math?

[Masum]

Assuming $a,b$ rational, such that $ab\neq q^2$ for a rational $q$, yes. Say, $\sqrt a+\sqrt b=s\Rightarrow \sqrt{ab}=\dfrac{s-a-b}{2}$.