# How do you rationalize the denominator and simplify (10-sqrt3)/(6+sqrt6)?

Mar 15, 2016

$\frac{60 - 10 \sqrt{6} - 6 \sqrt{3} + 3 \sqrt{2}}{30}$

#### Explanation:

$1$. Start by multiplying the numerator and denominator by the conjugate of the fraction's denominator, $6 - \sqrt{6}$.

$\frac{10 - \sqrt{3}}{6 + \sqrt{6}}$

$= \frac{10 - \sqrt{3}}{6 + \sqrt{6}} \left(\frac{6 - \sqrt{6}}{6 - \sqrt{6}}\right)$

$2$. Simplify the numerator.

$= \frac{60 - 10 \sqrt{6} - 6 \sqrt{3} + \sqrt{18}}{6 + \sqrt{6}} \left(\frac{1}{6 - \sqrt{6}}\right)$

$= \frac{60 - 10 \sqrt{6} - 6 \sqrt{3} + 3 \sqrt{2}}{6 + \sqrt{6}} \left(\frac{1}{6 - \sqrt{6}}\right)$

$3$. Simplify the denominator. Note that it contains a difference of squares $\left(\textcolor{red}{{a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)}\right)$.

$= \frac{60 - 10 \sqrt{6} - 6 \sqrt{3} + 3 \sqrt{2}}{36 - 6}$

$= \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \frac{60 - 10 \sqrt{6} - 6 \sqrt{3} + 3 \sqrt{2}}{30} \textcolor{w h i t e}{\frac{a}{a}} |}}}$