Power operations in $K$-theory completed at a prime

We describe the action of power operations on the $p$-completed cooperation algebras $K_0^{\vee }K=K_0(K){\text{sphat}}_p$ for $K$-theory at a prime~$p$, and $K_0^{\vee }KO=K_0(KO){\text{sphat}}_2$.