# How do you simplify (2sqrt5)/ (sqrt2 +5)?

Apr 9, 2015

To simplify the expression, we need to make it's Denominator RATIONAL. And to do that, we multiply the numerator and the denominator with it's CONJUGATE.

The denominator is $\sqrt{2} + 5$ which is the same as $5 + \sqrt{2}$

and it's CONJUGATE is $5 - \sqrt{2}$

$\frac{2 \sqrt{5}}{5 + \sqrt{2}} \cdot \frac{5 - \sqrt{2}}{5 - \sqrt{2}}$ (This keeps the Ratio unchanged)

$= \frac{\left(2 \sqrt{5}\right) \cdot \left(5 - \sqrt{2}\right)}{{5}^{2} - {\left(\sqrt{2}\right)}^{2}}$

$= \frac{\left(2 \sqrt{5}\right) \cdot \left(5 - \sqrt{2}\right)}{25 - 2}$

$= \frac{\left(2 \sqrt{5}\right) \cdot \left(5 - \sqrt{2}\right)}{23}$

$= \frac{10 \sqrt{5} - 2 \sqrt{10}}{23}$

As the denominator 23 is RATIONAL, we have successfully simplified the expression.