Just a little queations really
$\displaystyle \sqrt{50} = 5\times\sqrt{2}$
Now i can prove that $\displaystyle \sqrt{2}$ is irrational, can i just say that $\displaystyle 5\times\sqrt{2}$ is irrational with no extra proof?
I'd be convinced, but it is perhaps even more obvious if you do things the other way around. If $\displaystyle \surd 50$ were rational then you would have $\displaystyle \surd 2$ = $\displaystyle \frac{\surd 50}{5}$ so that then $\displaystyle \surd 2$ would be rational as well.