  We develop new tools leading, for each integer $n\ge 4$, to a significantly
improved upper bound for the uniform exponent of rational approximation
$\widehat{\lambda}_n(\xi)$ to successive powers $1,\xi,\dots,\xi^n$ of a given
real transcendental number $\xi$. As an application, we obtain a refined lower
bound for the exponent of approximation to $\xi$ by algebraic integers of
degree at most $n+1$. The new lower bound is $n/2+a\sqrt{n}+4/3$ with
$a=(1-\log(2))/2\simeq 0.153$, instead of the current $n/2+\mathcal{O}(1)$.
