# How do you simplify ((sqrt7) - (sqrt 2))/((sqrt7) + (sqrt 2))?

Apr 9, 2015

For this kind of a question you should multiply the nominator and the denominator with the demonator but with the minus sign. (forgive me for my bad english :). What i mean is, this is what you should do;

$\frac{\left(\sqrt{7}\right) - \left(\sqrt{2}\right)}{\left(\sqrt{7}\right) + \left(\sqrt{2}\right)}$ since there is $\left(\left(\sqrt{7}\right) + \left(\sqrt{2}\right)\right)$ in the denomintor you should multiply the whole equation with $\left(\left(\sqrt{7}\right) - \left(\sqrt{2}\right)\right)$

The reason we do this to reach this result in the denominator part;
$\left(\left(\sqrt{7}\right) + \left(\sqrt{2}\right)\right)$ . $\left(\left(\sqrt{7}\right) - \left(\sqrt{2}\right)\right)$ = $7 - 2$ = $5$
I came to that from this formula;
$\left(x + y\right) . \left(x - y\right) = {x}^{2} - {y}^{2}$ $\implies$ This is a formula that should be memorized in order to solve this kind of question.

Also you have to know this too;
${\left(x - y\right)}^{2} = {x}^{2} - 2 x y + {y}^{2}$

So if i solve the entire question the result will be;
$\frac{\left(\sqrt{7}\right) - \left(\sqrt{2}\right)}{\left(\sqrt{7}\right) + \left(\sqrt{2}\right)}$ = $\frac{\left(\sqrt{7}\right) - \left(\sqrt{2}\right)}{\left(\sqrt{7}\right) + \left(\sqrt{2}\right)} \cdot \frac{\left(\sqrt{7}\right) - \left(\sqrt{2}\right)}{\left(\sqrt{7}\right) - \left(\sqrt{2}\right)}$ = ${\left(\left(\sqrt{7}\right) - \left(\sqrt{2}\right)\right)}^{2} / \left(7 - 2\right)$ = $\frac{7 - 2 \sqrt{14} + 2}{5}$ = $\frac{9 - 2 \sqrt{14}}{5}$ $\implies$ this is the simplest version i can write. I hope it helps.