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\sqrt{3 -x^2} and \sqrt{a - x^3} cannot be rational, if x, x^3 < 3

Source: 2002 Romania JBMO TST 4.1

May 31, 2020
rationalalgebraradicalirrational

Problem Statement

Let aa be an integer. Prove that for any real number x,x3<3x, x^3 < 3, both the numbers 3−x2\sqrt{3 -x^2} and a−x3\sqrt{a - x^3} cannot be rational.
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