TITLE: Proving/disproving that √7 - √2 is irrational
QUESTION [5 upvotes]: It's been proven that √7 and √2 are irrational.
However, I am not sure how to go about proving that √7 - √2. Is it an acceptable proof to just solve the equation which would prove/disprove the equation or as should the proof be done as a contrapositive, similar to how √7 and √2 are proven to be irrational. 
What would a valid proof/disproof of irrationality look like in this case?

REPLY [7 votes]: Suppose
$\sqrt{a}-\sqrt{b}
= r
$
is rational.
Squaring this,
$a+b-2\sqrt{ab} = r^2$,
so
$\sqrt{ab}$ is rational.
If $\sqrt{ab}$ is irrational,
this can not hold.
Therefore,
if $\sqrt{ab}$ is irrational,
so is
$\sqrt{a}-\sqrt{b}$.
Since $\sqrt{14}$ is irrational,
so is
$\sqrt{7}-\sqrt{2}$.
Note that
this works for
$\sqrt{a}+\sqrt{b}$
also.
Note 2:
There are many proofs here that
if $n$ is not a perfect square
then $\sqrt{n}$
is irrational.