# Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by cubing both sides and using $x = \sqrt[3]{2}$ for simplified typing.

Ramanujan established many such denesting of radicals such as $$\sqrt{\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}} = \sqrt[5]{1 + \sqrt[5]{2} + \sqrt[5]{8}} = \sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}}\tag {2}$$$$\sqrt[3]{\sqrt[5]{\frac{32}{5}} - \sqrt[5]{\frac{27}{5}}} = \sqrt[5]{\frac{1}{25}} + \sqrt[5]{\frac{3}{25}} - \sqrt[5]{\frac{9}{25}}\tag {3}$$$$\sqrt[4]{\frac{3 + 2\sqrt[4]{5}}{3 - 2\sqrt[4]{5}}} = \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1}\tag{4}$$$$\sqrt[\color{red}6]{7\sqrt[3]{20} - 19} = \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}}\tag{5}$$$$\sqrt[6]{4\sqrt[3]{\frac{2}{3}} - 5\sqrt[3]{\frac{1}{3}}} = \sqrt[3]{\frac{4}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{1}{9}}\tag{6}$$

$$\sqrt[8]{1+\sqrt{1-\left(\frac{-1+\sqrt{5}}{2}\right)^{24}}} = \frac{-1+\sqrt{5}}{2}\,\frac{1+\sqrt[4]{5}}{\sqrt{2}}\tag{7}$$

with the last one found in Ramanujan's Notebooks, Vol 5, p. 300. Most of these radical expressions are units (a unit is an algebraic integer $\alpha$ such that $\alpha\beta = 1$ where $\beta$ is another algebraic integer).

For me the only way to establish these identities is to raise each side of the equation to an appropriate power using brute force algebra and then check the equality. However for higher powers (for example equation $(2)$ above) this seems very difficult.

Is there any underlying structure in these powers of units which gives rise to such identities or these are mere strange cases which were noticed by Ramanujan who used to play with all sorts of numbers as a sort of hobby? I believe (though not certain) that perhaps Ramanujan did have some idea of such structure which leads to some really nice relationships between units and their powers. I wonder if there is any sound theory of such relationships which can be exploited to give many such identities between nested and denested radicals.

• Yes, there is an algorithm which essentially comes from Kummer theory. See this. I think I am going to bet that Ramanujan didn't know about Galois theory. If he did, he might have just discovered algebraic number theory from it's roots (ps : Ramanujan was indeed the founder of algebraic number theory, but in a different sense. He actually discovered the notion of modular forms, which were formalized decades later and used frequently in algebraic number theory) – Balarka Sen Jul 19 '14 at 12:02
• @BalarkaSen: the structure theorem mentioned in your link seems to be what i wanted. Need to study in more detail on this theorem. thanks – Paramanand Singh Jul 19 '14 at 12:42
• @GerryMyerson: thanks for those links to other questions. one of them contains the kummer's theory mentioned in Balarka Sen's comments. I should have put more effort in searching on MSE first before my post. – Paramanand Singh Jul 19 '14 at 12:44
• @ParamanandSingh: I corrected a small typo for eq. 5 (in red), to be consistent with this post – Tito Piezas III Apr 14 '15 at 3:04

We also have the following identity,

$$\sqrt[3]{m^3-n^3+6m^2n+3mn^2-3(m^2+mn+n^2)\sqrt[3]{mn(m+n)}}=\\ \sqrt[3]{m^2(m+n)}-\sqrt[3]{mn^2}-\sqrt[3]{(m+n)^2n}$$

For $m=n=1$ we get $(1)$.

For $m=4$ and $n=1$ we get $(5)$.

• +1 Very nice. Note that the LHS can be simplified as $$\sqrt[3]{(m-n)^3+9m^2n-3(m^2+mn+n^2)\sqrt[3]{mn(m+n)}}$$ – Tito Piezas III Mar 26 '16 at 4:57
• @Frank your calculation. – Michael Rozenberg May 9 '16 at 3:19
• Can I have a proof for this identity? A possible link or a book? – Pragyaditya Das Apr 19 '17 at 8:24
• @MichaelRozenberg Is there any algebraic proof for this? – Pragyaditya Das Apr 19 '17 at 8:25
• @Pragyaditya Das Yes, of course! Let $a$, $b$ and $c$ be real roots of the equation $x^3+px^2+qx+r=0$. Also Let $\sqrt[3]a+\sqrt[3]b+\sqrt[3]c=A$ and $\sqrt[3]{ab}+\sqrt[3]{bc}+\sqrt[3]{ca}=B$. Hence, try to find $p$, $q$ and $r$ such that $A^3=kAB+n$ and $B^3=mAB+l$ solve the equation $x^3=(kx+n)(mx+l)$. – Michael Rozenberg Apr 19 '17 at 10:17

I found a PDF, where the authors have working algorithms to denest nested radicals like that of Ramanujan's, without the use of Galois theory.