Entry 6

Ramanujan was a master at manipulating radicals. Some of his unusual ones are
$$\sqrt[3]{\sqrt[5]{\frac{32}{5}} - \sqrt[5]{\frac{27}{5}}} = \sqrt[5]{\frac{1}{25}} + \sqrt[5]{\frac{3}{25}} + \sqrt[5]{\frac{-9}{25}}\tag1$$ $$\sqrt{\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}} = - \sqrt[5]{\frac{1}{125}}+\sqrt[5]{\frac{2}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{16}{125}} \tag2$$ $$\sqrt[4]{\frac{3 + 2\sqrt[4]{5}}{3 - 2\sqrt[4]{5}}} = \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1}\tag3$$ $$\sqrt[8]{1+\sqrt{1-\left(\frac{-1+\sqrt{5}}{2}\right)^{24}}} = \frac{-1+\sqrt{5}}{2}\,\frac{1+\sqrt[4]{5}}{\sqrt{2}}\tag4$$ The pattern of \((2)\) is certainly suggestive, though I haven't seen any generalization of this. And \((4)\) is the \(8\)th root of an expression involving a \(24\)th power! Also, note that he only uses ratios of \(\sqrt{p^k}\) with the small primes \(p=2,3,5\). Either such simple relations are possible only for these, or Ramanujan found the low-hanging fruits and there are similar ones with prime \(p>5\).