Problem #381

Denesting
Public 02/10/17 7xp Math 90.0%

Denesting is the process of simplifying an expression with embedded $n^{th}$ roots to obtain an equivalent expression with a lesser level of roots.

Ramanujan was a master in this matter and provided some amazing examples:
$ \sqrt[3]{\sqrt[3]{2}-1} = (\sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{4}{9}})\times \frac{1}{3} $
$ \sqrt{\sqrt[3]5 - \sqrt[3]4} = (\sqrt[3]2 + \sqrt[3]{20}-\sqrt[3]{25})\times \frac{1}{3} $
$ \sqrt[4] { \frac{3+2\sqrt[4]5}{3-2\sqrt[4]5} } = ( 3 + \sqrt[4]5 +\sqrt{5} + \sqrt[4]{125}) \times \frac{1}{2} $

Can you denest the following expression : $ \sqrt{ \sqrt[3]{1404} - \sqrt[3]{875} } $ ?

The solution has the form $ (\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c}) \times \frac{p}{q} \textrm{ with } a \lt b \lt c \quad p, q \textrm{ coprime} $

Answer format : $ a,b,c,p,q $

[My timing: 35 sec]



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