Paper 4, Section I, E

Numbers and Sets | Part IA, 2018

Given nNn \in \mathbb{N}, show that n\sqrt{n} is either an integer or irrational.

Let α\alpha and β\beta be irrational numbers and qq be rational. Which of α+q,α+β,αβ,αq\alpha+q, \alpha+\beta, \alpha \beta, \alpha^{q} and αβ\alpha^{\beta} must be irrational? Justify your answers. [Hint: For the last part consider 22\sqrt{2}^{\sqrt{2}}.]

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