Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational 
Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational.

The sum of a rational and irrational number is always irrational, that much I know - thus, if $n$ is a perfect square, we are finished.
However, is it not possible that the sum of two irrational numbers be rational? If not, how would I prove this?
This is a homework question in my proofs course.

By a simple 1-line proof: $\,\Bbb Q(\sqrt n\! +\! \sqrt 2) = \Bbb Q(\sqrt n,\sqrt 2),\, $ so $\,\sqrt n\! +\! \sqrt 2\in\Bbb Q\,\Rightarrow\sqrt 2\in \Bbb Q\,\Rightarrow\!\Leftarrow$
