How prove this number $(a,b,n)=(1,1,4k)$ Let $a,b,n\in \mathbb{N}$, $\gcd(a,b)=1$, $\left(\sqrt{a}+i\sqrt{b}\right)^n\in \mathbb{R}$

Show that
$(a,b,n)=(1,1,4k)$ or $(1,\sqrt{3},3k)$ or $(\sqrt{3},1,6k)$ where $k\in \mathbb{N}$, $i=\sqrt{-1}$
Let $x=\sin^{-1}\frac{\sqrt{b}}{\sqrt{a+b}}$, then $\cos 2x=\frac{a-b}{a+b} \in \mathbb{Q}$. Since $nx=k\pi$, using the well known theorem (see for example http://yaroslavvb.com/papers/olmsted-rational.pdf), $\cos 2x=0, \pm 1, \pm \frac{1}{2}$.
Let $x=\sin^{-1}\frac{\sqrt{b}}{\sqrt{a+b}}$, then $\cos 2x=\frac{a-b}{a+b} \in \mathbb{Q}$. Since $nx=k\pi$, using the well known theorem (see for example http://yaroslavvb.com/papers/olmsted-rational.pdf), $\cos 2x=0, \pm 1, \pm \frac{1}{2}$.