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