Abstract The group of formal power series under substitution over 𝔽 p {\mathbb{F}_{p}} , the so-called Nottingham group, is a pro-p group so it is a metric space. The intertwining of these two structures is an important object of study. This paper is concerned with how the power map influences the distance of elements. Suppose that d ( f , g ) = d 1 {d(f,\,g)=d_{1}} while d ( f , 1 ) = d 2 ≥ d 1 {d(f,1)=d_{2}\geq d_{1}} . In this paper we provide a sharp bound for d ( f j , g j ) {d(f^{j},\,g^{j})} in terms of d 1 , d 2 {d_{1},\,d_{2}} and the exponent j. This bound confirms a conjecture by K. Keating.