Minimal polynomial in $\mathbb{Q} [x]$

Question

Let $0＜ \rho＜1$ be an irrational and there exists a positive integer $n$ such that $\rho^n \in \mathbb{Q}$. Let $r$ be the smallest positive integer such that $u=\rho^r\in \mathbb{Q}$.

Question: can we show that $f(x)=x^r-u$ is the minimal polynomial of $\rho$ in $\mathbb{Q} [x]$?