I am looking for a way to check whether $(x^n - 1)$ is an $nth$ power when $x$ is rational and $n\in \Bbb N$.
So far I found some solutions such as $(1.25^2 - 1)$ is $0.75^2$.
I also found I can write $(x^n - 1)$ as $$(x - 1)(x^{n-1}+x^{n-2}+...+x+1)$$ or with roots of unity as $$(x-1)(x-ζ_1)(x-ζ_2)...(x-ζ_{n-1}) \quad where \quad ζ_k = e^{2\pi ik/n}$$ Though the second method looked promising, I couldn't figure out any way through it. I have only basic knowledge in mathematics. I would be grateful for any help.