Faster way to calculate $\frac{\sqrt8+\sqrt{27}}{5-\sqrt6}-2(\sqrt[4]9-1)^{-1}$ What is the value of  $\cfrac{\sqrt8+\sqrt{27}}{5-\sqrt6}-2(\sqrt[4]9-1)^{-1}$ ?
$$1)1+\sqrt3\quad\quad\quad\quad\quad\quad2)-1+\sqrt2\quad\quad\quad\quad\quad\quad3)1-\sqrt2\quad\quad\quad\quad\quad\quad4)\sqrt2-2\sqrt3$$
It was one of the questions from timed exam (for example supposed to solve any question in average one minute).  but I have difficulty to calculate this expression fast. my approach is:
$$\left(\frac{2\sqrt2+3\sqrt3}{5-\sqrt6}\times\frac{5+\sqrt6}{5+\sqrt6}\right)-\frac2{\sqrt3-1}$$
By rationalizing the second fraction it is not hard to see it is $\sqrt3+1$. but I had difficulty to accurately calculate the first one in short time. Is it possible to evaluate it quicker?

$$\begin{align}
\frac{\sqrt{8}+\sqrt{27}}{5-\sqrt{6}} 
&= \frac{(\sqrt{8} + \sqrt{27})(5 + \sqrt{6})}{25-6} \\
&= \frac{5 \sqrt{8} + \sqrt{48} + 5\sqrt{27} + \sqrt{81(2)}}{19} \\
&= \frac{10\sqrt{2} + 4\sqrt{3} + 15\sqrt{3} + 9\sqrt{2}}{19} \\
&= \sqrt{2} + \sqrt{3}.
\end{align}$$
