Exponentiation

In mathematics, exponentiation, denoted bn, is an operation 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 b by itself n times:

b

n

=

b

×

b

×

⋯

×

b

×

b

⏟

n

times

.

{\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.}

The definition of exponentiation can be extended in a natural way to define

b

x

{\displaystyle b^{x}}

for any positive real number base

b

{\displaystyle b}

and any real number exponent

x

{\displaystyle x}

. A more involved definition supports arbitrary complex numbers as base and exponent. Other generalizations allow algebraic objects, such as square matrices, to be the base.

Exponentiation is a fundamental operation used throughout mathematics, science, and engineering.

Notation

This article uses the three common notations for multiplication,

x

×

y

{\displaystyle x\times y}

,

x

y

{\displaystyle xy}

, and

x

⋅

y

{\displaystyle x\cdot y}

, interchangeably.

The exponent is usually shown as a superscript to the right of the base as bn. Sometimes an up arrow or caret is used as the operator, e.g.

b

↑

n

{\displaystyle b\uparrow n}

or b^n.