From Förberedande kurs i matematik 1

We can get rid of both square roots in the denominator if we multiply the top and bottom of the fraction by the conjugate expression \displaystyle \left( a-b \right)\left( a+b \right)=a^{2}-b^{2}, and use the conjugate rule

with \displaystyle a=\sqrt{17} and \displaystyle b=\sqrt{13}. Both roots are squared away and we get

\displaystyle \begin{align} & \frac{1}{\sqrt{17}-\sqrt{13}}=\frac{1}{\sqrt{17}-\sqrt{13}}\centerdot \frac{\sqrt{17}+\sqrt{13}}{\sqrt{17}+\sqrt{13}} \\ & =\frac{\sqrt{17}+\sqrt{13}}{\left( \sqrt{17} \right)^{2}-\left( \sqrt{13} \right)^{2}}=\frac{\sqrt{17}+\sqrt{13}}{17-13}=\frac{\sqrt{17}+\sqrt{13}}{4}. \\ \end{align}

This expression cannot be simplified any further because neither \displaystyle \text{17} nor \displaystyle \text{13} contain any squares as factors.