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 \sqrt{17}+\sqrt{13}, and use the difference of two squares

\displaystyle (a-b)(a+b) = a^2-b^2

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}}\cdot\frac{\sqrt{17}+\sqrt{13}}{\sqrt{17}+\sqrt{13}}\\[5pt] &= \frac{\sqrt{17}+\sqrt{13}}{(\sqrt{17})^{2}-(\sqrt{13})^{2}}\\[5pt] &= \frac{\sqrt{17}+\sqrt{13}}{17-13}\\[5pt] &= \frac{\sqrt{17}+\sqrt{13}}{4}\,\textrm{.} \end{align}

This expression cannot be simplified any further because neither 17 nor 13 contain any squares as factors.