# How do you rationalize the denominator and simplify ( 3sqrt6 + 5sqrt2)/(4sqrt6 - 3sqrt2)?

Mar 23, 2018

$\frac{102 + 29 \sqrt{12}}{78}$

#### Explanation:

$\textcolor{w h i t e}{\times} \frac{3 \sqrt{6} + 5 \sqrt{2}}{4 \sqrt{6} - 3 \sqrt{2}}$

$= \frac{\left(3 \sqrt{6} + 5 \sqrt{2}\right) \left(4 \sqrt{6} + 3 \sqrt{2}\right)}{\left(4 \sqrt{6} - 3 \sqrt{2}\right) \left(4 \sqrt{6} + 3 \sqrt{2}\right)}$

[Multiply Both Numerator and Denominator by $\left(4 \sqrt{6} + 3 \sqrt{2}\right)$]

$= \frac{72 + 20 \sqrt{12} + 9 \sqrt{12} + 30}{{\left(4 \sqrt{6}\right)}^{2} - {\left(3 \sqrt{2}\right)}^{2}}$

$= \frac{102 + 29 \sqrt{12}}{96 - 18}$

$= \frac{102 + 29 \sqrt{12}}{78}$

Hence explained.