Non-transcendental number with powers being irrational

Question

Is there a real number $r$ such that all $r^n$ are irrational for all integers $n\ge1$ but it is not transcendental?

Answer 1

Yes: $1+\sqrt 2$, for instance. $$(1+\sqrt2)^n=1+2\binom n2+2^2\binom n 4+\dots+\biggl(\binom n1+2\binom n3+2^2\binom n5+\dotsm\biggr)\sqrt2.$$

Answer 2

A number is transcendental if it is not root of any polynomial having coefficients in $\mathbb Q$. A number such that $r^n$ is irrational $\forall n$ is only not a root of $x^n-q$, $\forall n \forall q\in \mathbb Q$, but it could be not transcendental anyway, by being root of a polynomial coprime with the last one.