\[\frac{18}{\sqrt{7} + 1} = \frac{18\left( \sqrt{7} - 1 \right)}{\left( \sqrt{7} + 1 \right)\left( \sqrt{7} - 1 \right)} =\]
\[= \frac{18\left( \sqrt{7} - 1 \right)}{7 - 1} = \frac{18\left( \sqrt{7} - 1 \right)}{6} =\]
\[= 3\left( \sqrt{7} - 1 \right)\]