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This question has already been asked before, but has failed to generate any sufficient response. A comment on the other post suggests the problem might be incorrect. I'd really appreciate some help.
Alternate post here.If $(a-b^2)b>0$, then $\sqrt[3]{a+\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}+\sqrt[3]{a-\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}$ is rational
A potential idea is perhaps generating a cubic whose cardano solutions are the above, but that still tenders the sum of conjucate radicals a real number, not necessarily a rational. Unless we can use some equivalent in Q[X].