<s>[INST] Proving that the number $\sqrt[3]{7 + \sqrt{50}} + \sqrt[3]{7 - 5\sqrt{2}}$ is rational I've been struggling to show that the number $\sqrt[3]{7 + \sqrt{50}} + \sqrt[3]{7 - 5\sqrt{2}}$ is rational. I would like to restructure it to prove it, but I can't find anything besides $\sqrt{50} =5 \sqrt{2}$. Could anybody give me some hints? Thanks in advance!
 [/INST] use this well known identity
$$(a+b)^3=a^3+b^3+3ab(a+b)$$,so we Let
$$x=\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}$$
then we have
$$x^3=14+3\sqrt[3]{49-50}\cdot x=-3x+14$$
so $$x=2$$
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