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 ?

we have $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number
and so is their product.(which you found)
Now, squaring $\sqrt[3]{a}+\sqrt[3]{b}$ we can get,
$\sqrt[2/3]{b}+\sqrt[2/3]{a}$ is also rational(using the fact that $\sqrt[3]{ab} \in \mathbb Q$
Similarly,squaring $\sqrt[2/3]{b}+\sqrt[2/3]{a}$
we can get, $\sqrt[4/3]{b}+\sqrt[4/3]{a}=a*\sqrt[3]{a}+b*\sqrt[3]{b} \in \mathbb Q$
Now,multiply  $\sqrt[3]{a}+\sqrt[3]{b}$ by a and subtract it from $a*\sqrt[3]{a}+b*\sqrt[3]{b}$
to get the desired result.
