Faster way to calculate $\frac{\sqrt8+\sqrt{27}}{5-\sqrt6}-2(\sqrt[4]9-1)^{-1}$

Question

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?

Answer 1

If you are very clever, it might occur to you that

$$\sqrt8+\sqrt{27}=(\sqrt2)^3+(\sqrt3)^3=(\sqrt2+\sqrt3)((\sqrt2)^2-\sqrt2\sqrt3+(\sqrt3)^2)=(\sqrt2+\sqrt3)(2-\sqrt6+3)$$

which lets you cancel out the $5-\sqrt6$ from the denominator. But that only occurred to me after the fact.

Answer 2

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