Interesting Surd Problem

Question

If $ x $, $ y $ and $ z $ are rational numbers such that $ \sqrt[3]{\sqrt[3]{2}-1} = \sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}$ then find $ x,y,z $

Answer 1

Firstly:

$${2}^{1/3}-1=(x^{1/3}+y^{1/3}+z^{1/3})^3=x+y+z+...=\frac mn+...$$

$${2}^{1/3}=\frac mn+1+...$$

Now you should show that in such case $m/n\equiv-1$, so $m+n=0$

Answer 2

According to Srinivasa Ramanujan, $\sqrt[3]{\sqrt[3]{2}-1} =\sqrt[3]{\frac{1}{9}} -\sqrt[3]{\frac{2}{9}} +\sqrt[3]{\frac{4}{9}}$.