# Suppose x is a nonzero real number such that both x5 and 20x + 19/x are rational numbers. Prove that x is a rational number.

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Suppose x is a nonzero real number such that both xand 20x + 19/x are rational numbers. Prove that x is a rational number.

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Since x5 is rational, we see that (20x)5 and (x/19)5 are rational numbers. But

is also rational. Thus T is a rational number and T  0. We conclude that 20x −(19/x) is a rational number. This combined with the given condition that 20x + (19/x) is rational shows 2 · 20 · x is rational. Therefore x is rational.