Someone asked a question about a proof:
Ramanujan found that
$$\begin{align*} & \sqrt[3]{(m^2+mn+n^2)\sqrt[3]{(m-n)(m+2n)(2m+n)}+3mn^2+n^3-m^3}\\ =&\sqrt[3]{\tfrac {(m-n)(m+2n)^2}9}-\sqrt[3]{\tfrac {(2m+n)(m-n)^2}9}+\sqrt[3]{\tfrac {(m+2n)(2m+n)^2}9} \end{align*}$$ for arbitrary $m$ and $n$.
Question: Is there a way to prove it?
The answer given essentially worked the RHS into the LHS, which completes the proof. I'm wondering specifically about the intuition that led Ramanujan (or anyone trying to denest $\sqrt[3]{\sqrt[3]{x} + y}$) to such a formula. Is it just another one of those "he thought of it in a dream" and then proved afterward, or is there an intuitive proof, starting point, assumption, etc. that can lead to the same conclusion?
Also, does it even matter? Does it suffice (as in enough information for someone first learning about this topic) to present the formula and then prove it similar to that of the answer given?
