structural:scholar_edition, linguistic:paraphrasing
Integer powers in algebra

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. The definition of x requires further the existence of a multiplicative identity.

An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a monoid. In such a monoid, exponentiation of an element x is defined inductively by

If n is a negative integer, x n {\displaystyle x^{n}} is defined only if x has a multiplicative inverse. In this case, the inverse of x is denoted x, and x is defined as (x − 1) − n. {\displaystyle \left(x^{-1}\right)^{-n}.}

Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:

These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and endomorphisms of any mathematical structure.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if f is a real function whose valued can be multiplied, f n {\displaystyle f^{n}} denotes the exponentiation with respect of multiplication, and f ∘ n {\displaystyle f^{\circ n}} may denote exponentiation with respect of function composition. That is,

and

Commonly, (f n) (x) {\displaystyle (f^{n})(x)} is denoted f (x) n, {\displaystyle f(x)^{n},} while (f ∘ n) (x) {\displaystyle (f^{\circ n})(x)} is denoted f n (x). {\displaystyle f^{n}(x).}