rationality testing - Proof that $2sqrt2 + sqrt7$ is an irrational number

Prove that $2\sqrt2 + \sqrt7$ is an irrational number. I am trying to use contradiction to show that this is irrational. Also I am using the fact that $ 2\sqrt2 + \sqrt7 = \frac{1}{2\sqrt2 - \sqrt7}$.

Assume $2\sqrt2 + \sqrt7 $ is rational. Then

$2\sqrt2 + \sqrt7 = \frac{m}{n}$

and

$2\sqrt2 - \sqrt7 = \frac{n}{m}$

if I add them together I get

$ 4\sqrt2 = \frac{m^2+n^2}{mn} $

then we can divide the 4 on both sides and label the new numerator and denominator as $ x , y$ then to prove $\sqrt2$ is irrational is trivial. I am not sure if this is the correct way of proving this. Like why does it work to add it's reciprocal? Is it because we are assuming it is rational and if it is rational so is its reciprocal? And rational numbers are closed under addition?

Answer

The proof works because if your number is a rational $\frac mn$, then its reciprocal, $\frac nm$ is also rational: it is the quotient of two integers. (The only way that it wouldn't be rational is in the case $m=0$, but that would imply that $2\sqrt2 = -\sqrt7$, which is clearly false.)

Similarly, when you add $\frac mn$ to $\frac nm$, you get another rational number, $\frac{m^2+n^2}{mn}$.