Let $\displaystyle a,b \in\mathbb{N}^+$ such that $\displaystyle \sqrt{a^2+b^2} \in \mathbb{N}$.

Show that $\displaystyle \forall n\in\mathbb{N}^+\; (a+ib)^n \notin \mathbb{R}$

Where $\displaystyle i \in \mathbb{C}$ is the imaginary unit, and $\displaystyle \mathbb{N}^+,\mathbb{R, C}$ are the sets of all positive integers, reals and complex numbers respectively.