A problem of Mahler on farctional parts of powers of an algebraic number is solved, namely a classification is provided of the algebraic numbers $\alpha$ such that the fractional powers of $\alpha^n$ tends to zero exponentially on a sequence of integers.
A problem of Mendes France is solved, by proving that the period length of the continued fraction of the powers of a quadratic irrational tends to infinity apart trivial cases.
