<s>[INST] Prove that $\sqrt{2} +\sqrt{n}$ is irrational Suppose $\sqrt{2} + \sqrt{n}$ is rational,
so it can be represented like so: $\sqrt{2} + \sqrt{n} = {x \over y}$
then  $\sqrt{2} = {{x \over y} - \sqrt{n}}$
so $\sqrt{2} = {x - y\sqrt{n}\over y}$
but $\sqrt{2}$ is irrational, which leads to a contradiction.
Does this proof make sense?
edit: fixed some mistakes
 [/INST] The sum or product of rational numbers is rational.
If $\sqrt 2 + \sqrt n$ is rational, then so is $\frac{2 - n}{\sqrt 2 + \sqrt n} = \sqrt 2 - \sqrt n$.
Then $\frac{\left(\sqrt 2 + \sqrt n\right) + \left(\sqrt 2 - \sqrt n\right)}{2} = \sqrt 2$ is rational.
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