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Sum is rational implies numbers are rational

Source: India TST 2016 Day 2 Problem 1

July 22, 2016
algebrairrational number

Problem Statement

Suppose α,β\alpha, \beta are two positive rational numbers. Assume for some positive integers m,nm,n, it is known that α1n+β1m\alpha^{\frac 1n}+\beta^{\frac 1m} is a rational number. Prove that each of α1n\alpha^{\frac 1n} and β1m\beta^{\frac 1m} is a rational number.
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