  About fifty years ago Mahler proved that if $\alpha>1$ is rational but not an
integer and if $0<l<1$ then the fractional part of $\alpha^n$ is $>l^n$ apart
from a finite set of integers $n$ depending on $\alpha$ and $l$. Answering
completely a question of Mahler we show that the same conclusion holds for all
algebraic numbers which are not $d$-th roots of Pisot numbers. By related
methods, we also answer a question by Mendes France, characterizing completely
the quadratic irrationals $\alpha$ such that the continued fraction of
$\alpha^n$ has period length tending to infinity.
