Let $$a = \frac{1}{{2 - \sqrt 3 }} + \frac{1}{{3 - \sqrt 8 }} + \frac{1}{{4 - \sqrt {15} }}$$       then we have

$$\eqalign{ & a = \frac{1}{{2 - \sqrt 3 }} + \frac{1}{{3 - \sqrt 8 }} + \frac{1}{{4 - \sqrt {15} }} \cr & \Rightarrow \frac{1}{{2 - \sqrt 3 }} \times \frac{{2 + \sqrt 3 }}{{2 + \sqrt 3 }} + \frac{1}{{3 - \sqrt 8 }} \times \frac{{3 + \sqrt 8 }}{{3 + \sqrt 8 }} + \frac{1}{{4 - \sqrt {15} }} \times \frac{{4 + \sqrt {15} }}{{4 + \sqrt {15} }} \cr & \Rightarrow \frac{{2 + \sqrt 3 }}{{4 - 3}} + \frac{{3 + \sqrt 8 }}{{9 - 8}} + \frac{{4 + \sqrt {15} }}{{16 - 15}} \cr & \Rightarrow 2 + \sqrt 3 + 3 + \sqrt 8 + 4 + \sqrt {15} \cr & \Rightarrow 9 + \sqrt 3 + 2\sqrt 2 + \sqrt {15} \cr & a = 9 < 9 + \sqrt 3 + 2\sqrt 2 + \sqrt {15} < 18 \cr & \sqrt 3 = 1.73,\,\sqrt 2 = 1.41,\,\sqrt {15} = 3.9 \cr & \Rightarrow 9 < 9 + 1.73 + \left( {2 \times 1.41} \right) + 3.9 = 17.4 < 18 \cr} $$