How do you rationalize (2sqrt3-sqrt2)/ (5sqrt2+sqrt3)?

May 10, 2015

Given $\frac{2 \sqrt{3} - \sqrt{2}}{5 \sqrt{2} + \sqrt{3}}$, try multiplying both the top and the bottom by $\left(5 \sqrt{2} - \sqrt{3}\right)$

The numerator:

$\left(2 \sqrt{3} - \sqrt{2}\right) \left(5 \sqrt{2} - \sqrt{3}\right)$

$= - 2 \sqrt{3} \sqrt{3} + 9 \sqrt{2} \sqrt{3} - 5 \sqrt{2} \sqrt{2}$

$= - 2 \cdot 3 + 9 \sqrt{2 \cdot 3} - 5 \cdot 2$

$= 9 \sqrt{6} - 16$

The denominator:

$\left(5 \sqrt{2} + \sqrt{3}\right) \left(5 \sqrt{2} - \sqrt{3}\right)$

$= 25 \sqrt{2} \sqrt{2} - \sqrt{3} \sqrt{3}$

$= 50 - 3$

$= 47$.

So $\frac{2 \sqrt{3} - \sqrt{2}}{5 \sqrt{2} + \sqrt{3}} = \frac{9 \sqrt{6} - 16}{47}$.