\[\frac{5}{\sqrt{13} - \sqrt{3}} =\]
\[= \frac{5\left( \sqrt{13} + \sqrt{3} \right)}{\left( \sqrt{13} - \sqrt{3} \right)\left( \sqrt{13} + \sqrt{3} \right)} =\]
\[= \frac{5\left( \sqrt{13} + \sqrt{3} \right)}{13 - 3} =\]
\[= \frac{5\left( \sqrt{13} + \sqrt{3} \right)}{10} = \frac{\sqrt{13} + \sqrt{3}}{2}.\]