Answer
$\dfrac{3\sqrt{5} + 3\sqrt{15} - 2\sqrt 3 - 6}{33}$
Work Step by Step
First it can be easily spotted that the denominator radicals could be simplified by the difference of two squares, therefore the numerator and denominator will be multiplied by $\left(3\sqrt 5 - 2\sqrt 3\right)$ to obtain:
$$=\dfrac{(3\sqrt 5 - 2\sqrt 3).(1 + \sqrt 3)}{(3\sqrt 5 - 2\sqrt 3).(3\sqrt 5 + 2\sqrt 3)}$$
By expanding the brackets the denominaotr becomes:
\begin{align*}&=\dfrac{(3\sqrt 5 - 2\sqrt 3)(1 + \sqrt 3)}{(3\sqrt 5)^{2} - (2\sqrt 3)^2}\\
\\&=\dfrac{(3\sqrt 5 - 2\sqrt 3)(1 + \sqrt 3)}{9\cdot5-4\cdot3}\\
\\&=\dfrac{(3\sqrt 5 - 2\sqrt 3)(1 + \sqrt 3)}{9\cdot5-4\cdot3}
\\&=\dfrac{(3\sqrt 5 - 2\sqrt 3)(1 + \sqrt 3)}{45-12}
\\&=\dfrac{(3\sqrt 5 - 2\sqrt 3)(1 + \sqrt 3)}{33}
\end{align*}
The denominator now is free of radicals and is in the simplest form, the next step is to expand and simplify the numerator.
\begin{align*}
&=\frac{3\sqrt 5(1+\sqrt3)-2\sqrt3(1+\sqrt3)}{(9.5) - (4.3)}\\
\\&=\frac{3\sqrt 5 + 3\sqrt{15} - 2\sqrt 3 - 2\sqrt 9}{(9.5) - (4.3)}\\
\\&=\frac{3\sqrt 5 + 3\sqrt{15} - 2\sqrt 3 - 2(3)}{33}\\
\\&=\frac{3\sqrt 5 + 3\sqrt{15} - 2\sqrt 3 - 6}{33}
\end{align*}
The final answer then becomes:
$$=\frac{3\sqrt 5 + 3\sqrt{15} - 2\sqrt 3 - 6}{33}$$
