Hello. I'd like to ask if there is any possibility that
sqrt(7) - sqrt(x)
can give a rational number?
Please, give the proof.
You do understand this wastes people's time, if you don't specify the question clearly and we end up answering the wrong question?
Let $\displaystyle a,b\in\mathbb{Z},b\ne0$
$\displaystyle \sqrt{7}-\sqrt{x}=\frac{a}{b}$
$\displaystyle \sqrt{7}-\frac{a}{b}=\sqrt{x}$
$\displaystyle x=(\sqrt{7}-\frac{a}{b})^2$
$\displaystyle x = 7-\left(\frac{2a}{b}\right)\sqrt{7}+\frac{a^2}{b^2}$
With a little more work we see that the only way to make x rational is to let a = 0.