MHB Calculate $r,s,t$ in $\displaystyle \sqrt[3]{\sqrt[3]{2} - 1}$

[juantheron]

Calculate number $r,s,t$ in $\displaystyle \sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{r} + \sqrt[3]{s} + \sqrt[3]{t}$

where $r,s,t \in Q$

 

[Sudharaka]

Calculate number $r,s,t$ in $\displaystyle \sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{r} + \sqrt[3]{s} + \sqrt[3]{t}$

where $r,s,t \in Q$

Hi jacks, :)

Can you please tell me where you encountered this problem? Is this a equation that you obtained as a consequence of your research/project, or is it a question from a book?

Kind Regards,
Sudharaka.

 

[Opalg]

Calculate number $r,s,t$ in $\displaystyle \sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{r} + \sqrt[3]{s} + \sqrt[3]{t}$

where $r,s,t \in Q$

I found that $ \sqrt[3]{\mathstrut\sqrt[3]{2} - 1} = \sqrt[3]{\frac19} + \sqrt[3]{-\frac29} + \sqrt[3]{\frac49}$. I did this more or less by trial and error, so the method is not very revealing. But you can verify that it is correct by cubing both sides.