<s>[INST] INMO : Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.
My first response was, isn't is obvious? but then I tried..
if $a=b$ then we prove it easily . So suppose $a\ne b$.
Let $s=\sqrt[3]{a} + \sqrt[3]{b}$.
then $s^3 = a + b + 3 s \sqrt[3]{a} \sqrt[3]{b}$
then the product $p=\sqrt[3]{a} \sqrt[3]{b}$ is a rational number.
But what should I do next ?
 [/INST] From your work, we can write $$\frac{p}{\sqrt[3]{a}} + \sqrt[3]{a} = s,$$ so $$\sqrt[3]{a} = \frac{s\pm \sqrt{s^2-4p}}{2}.$$
Cube both sides and solve for $\sqrt{s^2-4p}$ (noting $s^2-p \neq 0$) to show it's rational, which then shows $\sqrt[3]{a}$ is rational.
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