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

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