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Exponentiation (xⁿ) ○◂｜Definition｜1st｜20251119205401-00-⌔

Exponentiation - Wikipedia

Exponentiation

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In mathematics, exponentiation, denoted b${}^n$, is an operation involving two numbers: the base, b, and the exponent or power, n.¹ When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b${}^n$ is the product of multiplying n bases:¹

$b^n=\underset{n\text{ times}}{\underbrace{b\times b\times \cdots \times b\times b}}.$

In particular, $b^1=b$.

The exponent is usually shown as a superscript to the right of the base as b${}^n$ or in computer code as b^n. This binary operation is often read as “b to the power n”; it may also be referred to as “b raised to the n th power”, “the n th power of b”,² or, most briefly, “b to the n”.

The above definition of $b^n$ immediately implies several properties, in particular the multiplication rule:³

$\begin{array}{rl}{b}^n\times b^m& =\underset{n\text{ times}}{\underbrace{b\times \cdots \times b}}\times \underset{m\text{ times}}{\underbrace{b\times \cdots \times b}}\\ & =\underset{n+m\text{ times}}{\underbrace{b\times \cdots \times b}}=b^{n+m}.\end{array}$

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add.

Exponentiation can also be extended to powers that are not positive integers. When b is non-zero, the definition

$b^0=1$

is compatible with the multiplication rule: $b^0\times b^n=b^{0+n}=b^n$. A similar argument suggests the definition

$b^{-n}=1/b^n,$

for negative integer powers, and in particular $b^{-1}=\frac{1}{b}$ for any nonzero number b, and also the definition

$b^{n/m}=\sqrt[m]{b^n}$

for fractional powers (when m and n are both integers). For example, $b^{1/2}\times b^{1/2}=b^{1/2+1/2}=b^1=b$, meaning $(b^{1/2}{)}^2=b$, which is the definition of square root: $b^{1/2}=\sqrt{b}$.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define $b^x$ for any positive real base $b$ and any real number exponent $x$. More involved definitions allow complex base and exponent, as well as certain types of matrices as base or exponent.

Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

Printed 2026-06-28.

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Footnotes

1. Nykamp, Duane. “Basic rules for exponentiation”. Math Insight. Retrieved 2020-08-27. ↩ ↩²

2. Weisstein, Eric W. “Power”. MathWorld. Retrieved 2020-08-27. ↩

3. There are three common notations for multiplication: $x\times y$ is most commonly used for explicit numbers and at a very elementary level; $xy$ is most common when variables are used; $x\cdot y$ is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing. ↩

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