How to arrive at Ramanujan nested radical identity

Solution 1:

Hint $\ $ Multiplying by $\,\sqrt[3]{12}\,$ the RHS becomes $\, \sqrt[3]{20}- 2,\,$ and

$$(\sqrt[3]{20}- 2)^6\, =\, 144\, (7 \sqrt[3]{20} -19)$$

One can prove by Galois theory a structure theorem which essentially implies that all radical denestings can similarly be "normalised" to such a "trivial" denesting. This forms the foundation for effective algorithms to compute such denestings. See my post here and here for much more, including examples and literature references.

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