= Hall-Petrescu formula
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The Hall-Petrescu collection formula corrects the commutative rule $(xy)^n=x^ny^n$ by ordered products of higher <group commutators>. Its simplest useful form occurs in a <two-step nilpotent group>. With $[x,y]=x^{-1}y^{-1}xy$,
$$
(xy)^n=x^ny^n[y,x]^{\binom n2}\qquad(n\geq0).
$$
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 $\binom{n+1}{2}=\binom n2+n$ prove the formula.
For an odd <prime number> $p$, the coefficient $\binom p2$ is divisible by $p$. Thus if the <commutator subgroup> in this two-step setting has exponent dividing $p$, the $p$th power map satisfies $(xy)^p=x^py^p$. 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>.
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