\[\frac{18}{8 - 2\sqrt{7}} - \frac{18}{8 + 2\sqrt{7}} =\]
\[= \frac{18 \cdot \left( 8 + 2\sqrt{7} - 8 + 2\sqrt{7} \right)}{64 - 4 \cdot 7} =\]
\[= \frac{18 \cdot 4\sqrt{7}}{64 - 28} = \frac{72\sqrt{7}}{36} = 2\sqrt{7}\]