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The Hall-Petrescu collection formula corrects the commutative rule $(xy{)}^n=x^n{y}^n$ by ordered products of higher group commutators. Its simplest useful form occurs in a two-step nilpotent group. With $[x,y]=x^{-1}{y}^{-1}xy$,Here $[y,x]$ is central. The relation $yx=xy[y,x]$ shows that moving the new $x$ past $n$ copies of $y$ adds $n$ to its exponent; mathematical induction and $\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)+n$ prove the formula.

For an odd prime number $p$, the coefficient $\left(\genfrac{}{}{0ex}{}{p}{2}\right)$ is divisible by $p$. Thus if the commutator subgroup in this two-step setting has exponent dividing $p$, the $p$th power map satisfies $(xy{)}^p=x^p{y}^p$. For $p=2$ the correction generally remains. In higher nilpotency class, further commutators enter from deeper terms of the lower central series; their collection coefficients, rather than an assumption of commutativity, control power calculations in powerful p-groups and p-valued groups.