Exponentiation

Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases:

In that case, bⁿ is called the n-th power of b, or b raised to the power n.

The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the square of b (b²) or b squared; the exponent 3 (or 3rd power) is called the cube of b (b³) or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b.

When n is a negative integer and b is not zero, bⁿ is naturally defined as 1/b⁻ⁿ, preserving the property bⁿ × b^m = b^{n + m}.

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.