We describe the action of power operations on the p-completed cooperation
algebras K^\vee_0 K = K_0(K)\sphat_p for K-theory at a prime~p, and
K^\vee_0 KO = K_0(KO)\sphat_2.Comment: Version 6: final update, to appear in special issue of the Tbilisi
Mathematical Journal on Homotopy Theory, Spectra, and Structured Ring Spectr