Q:

Non-transcendental number with powers being irrational

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

A:

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

