Since x^{5} 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.