  We argue that supersymmetric higher-dimension operators in the effective
actions of M-theory and IIB string theory do not affect the maximally
supersymmetric vacua: $adS_4\times S^7$ and $adS_7\times S^4$ in M-theory and
$adS_5\times S^5$ in IIB string theory. All these vacua are described in
superspace by a fixed point with all components of supertorsion and
supercurvature being supercovariantly constant. This follows from 32 unbroken
supersymmetries and allows us to prove that such vacua are exact.
