An unusual identity – math.stackexchange.com

An unusual identity – math.stackexchange.com

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Prove that $$\left(\frac{\sqrt[3]{81+33\sqrt6}+\sqrt[3]{81-33\sqrt6}}{6}\right)^m=\left(\frac{\sqrt[3]{41+29\sqrt2}+\sqrt[3]{41-29\sqrt2}}{2}\right)^n$$ for all natural integers $m,n$.

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