This paper is concerned with the determination of the proof-strength of the power set axiom relative to axiom systems for Feferman's explicit mathematics. As conjectured by Feferman, we obtain that the presence of the power set axiom does not increase proof-strength. Results are achieved by reducing the systems including the power set axiom to subsystems of classical analysis. In those cases where only the induction axiom is available, we make use of the technique of asymmetrical interpretations.