In abstract algebra, power associativity is a property of a binary operation which is a weak form of associativity.
An algebra (or more generally a magma) is said to be powerassociative if the subalgebra generated by any element is associative. Concretely, this means that if an element x is multiplied by itself several times, it doesn't matter in which order the multiplications are carried out, so for instance x(x(xx)) = (x(xx))x = (xx)(xx). This is stronger than merely saying that (xx)x = x(xx) for every x in the algebra, but weaker than associativity, which requires that (xy)z = x(yz) for every x, y, and z in the algebra.
Every associative algebra is obviously powerassociative, but so are all other alternative algebras (like the octonions, which are nonassociative) and even some nonalternative algebras like the sedenions.
Exponentiation to the power of any natural number other than zero can be defined consistently whenever multiplication is powerassociative. For example, there is no ambiguity as to whether x^{3} should be defined as (xx)x or as x(xx), since these are equal. Exponentiation to the power of zero can also be defined if the operation has an identity element, so the existence of identity elements becomes especially useful in powerassociative contexts.
A nice substitution law holds for real powerassociative algebras with unit, which basically asserts that multiplication of polynomials works as expected. For f a real polynomial in x, and for any a in such an algebra define f(a) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that (fg) (a) = f(a)g(a).
References
 Albert, A. Adrian (1948), "Powerassociative rings", Transactions of the American Mathematical Society 64: 552–593, MR0027750, ISSN 00029947, http://www.jstor.org/stable/1990399
 R.D. Schafer, An introduction to nonassociative algebras, Dover, 1995, ISBN 0486688135. Chap.V, pp.128–148.
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