$$ a,b \in  \mathbb{Q} $$ $$ \sqrt{a} \notin \mathbb{Q}  ~\text{and}~ \sqrt{b} \notin \mathbb{Q} \implies \sqrt{a} + \sqrt{b} \notin \mathbb{Q}    $$  I'm not really sure on how to solve this, however, I know it's true, it's an implication of a theorem,  how can I prove this directly?
I need to prove this equivalence   Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ if and only if $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$  I have already proved the reciprocal (the obvious part) but I am having difficulties with the other part  Please any hint will be appreciated.  Thanks in advance.
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