Question

Избавиться от иррациональности в знаменателе дроби

Answer 1

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Answer 2

$\displaystyle \tt \frac{1}{\sqrt{2}-\sqrt{3}+\sqrt{5}}=\frac{\sqrt{2}-\sqrt{3}-\sqrt{5}}{(\sqrt{2}-\sqrt{3}+\sqrt{5})(\sqrt{2}-\sqrt{3}-\sqrt{5})}=\\\\\\\ =\frac{\sqrt{2}-\sqrt{3}-\sqrt{5}}{(\sqrt{2}-\sqrt{3})^{2}-(\sqrt{5})^{2}}=\frac{\sqrt{2}-\sqrt{3}-\sqrt{5}}{2-2\sqrt{6}+3-5}=-\frac{\sqrt{2}-\sqrt{3}-\sqrt{5}}{2\sqrt{6}}=\\\\\\=-\frac{\sqrt{6}\cdot (\sqrt{2}-\sqrt{3}-\sqrt{5})}{12}=-\frac{\sqrt{12}-\sqrt{18}-\sqrt{30}}{12}=\frac{\sqrt{30}+3\sqrt{2}-2\sqrt{3}}{12};$