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Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are rational.

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You can find a pretty nice solution to a generalized form here if you have an art of problem solving account : – dinoboy Jan 22 '13 at 21:31
Hint $\ $ See this answer to a closely related question. – Math Gems Jan 22 '13 at 21:56

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