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

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The problem involves calculating the numbers \( r, s, t \) such that \( \sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{r} + \sqrt[3]{s} + \sqrt[3]{t} \) with \( r, s, t \in \mathbb{Q} \). A participant confirmed the solution as \( r = \frac{1}{9}, s = -\frac{2}{9}, t = \frac{4}{9} \). This was found through trial and error, and the correctness can be verified by cubing both sides of the equation. The discussion also touches on the origin of the problem, with inquiries about its source. The conversation emphasizes the importance of verifying solutions in mathematical problems.
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Calculate number $r,s,t$ in $\displaystyle \sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{r} + \sqrt[3]{s} + \sqrt[3]{t}$

where $r,s,t \in Q$
 
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jacks said:
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.
 
jacks said:
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.
 

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