Lösung 3.1:7a

\frac{1}{\sqrt{6}-\sqrt{5}} &= \frac{1}{\sqrt{6}-\sqrt{5}}\cdot \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}\\[5pt] &= \frac{\sqrt{6}+\sqrt{5}}{(\sqrt{6})^{2}-(\sqrt{5})^{2}}\\[5pt] &= \frac{\sqrt{6}+\sqrt{5}}{6-5}\\[5pt] &= \sqrt{6}+\sqrt{5}\,,\\[10pt] \frac{1}{\sqrt{7}-\sqrt{6}} &= \frac{1}{\sqrt{7}-\sqrt{6}}\cdot \frac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}}\\[5pt] &= \frac{\sqrt{7}+\sqrt{6}}{(\sqrt{7})^{2}-(\sqrt{6})^{2}}\\[5pt] &= \frac{\sqrt{7}+\sqrt{6}}{7-6}\\[5pt] &= \sqrt{7}+\sqrt{6}\,\textrm{.} \end{align}

\frac{1}{\sqrt{6}-\sqrt{5}}-\frac{1}{\sqrt{7}-\sqrt{6}} &= \sqrt{6}+\sqrt{5}-(\sqrt{7}+\sqrt{6})\\[5pt] &= \sqrt{6}+\sqrt{5}-\sqrt{7}-\sqrt{6}\\[5pt] &= \sqrt{5}-\sqrt{7}\,\textrm{.} \end{align}