**Exponentiation** is a mathematical operation \(a^b = a\) multiplied by itself \(b\) times. For example, \(3^3 = 3 \times 3 \times 3 = 27\). It is ubiquitous in modern mathematics. \(a^b\) is typically pronounced "\(a\) to the \(b\)th power" or \(a\) to the \(b\)th." \(a\) is called the base, and \(b\) the **exponent**. The adjective form of exponentiation is **exponential**.

In googology, it is the third hyper operator. When repeated, it forms tetration.

In the fast-growing hierarchy, \(f_2(n) = n \times 2^n\) corresponds to exponential growth rate.

## Contents

## Definition

For a real number \(a\) and a non-negative integer \(b\), exponentiation has the following definition:

\[a^b := \prod_{i = 1}^{b} a\]

More precisely, it is defined in the following recursive way:

\[a^b := \left\{ \begin{array}{ll} 1 & (b = 0) \\ a^{b-1} \times a & (b > 0) \end{array} \right.\]

For a positive real number \(a\) and a real number \(b\), \(a^b\) is defined as \(e^{b \ln a}\), where \(e^x\) is the exponential function and \(\ln\) is the natural logarithm, which are defined like so:

\[e^x := \sum_{i = 0}^{\infty} \frac{x^i}{i!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\] \[\ln x := \int_1^x \frac{dt}{t}\]

Here \(n!\) denotes the factorial of \(n\). This allows the definition to be expanded to non-integer exponents. Despite the \(x^i\) terms in the definition of \(e^x\), the definition is not circular, because we defined \(a^b\) for the case where \(b\) is a non-negative integer in a distinct way. Moreover, the values of \(a^b\) with respect to those two definitions coincide with each other for any positive real number \(a\) and any non-negative integer \(b\).

Similarly, the exponentiation is extended to elements in other complete topological rings such as complex numbers, \(p\)-adic numbers, and matrices with appropriate coefficients. This is a special property of exponentiation, and few analogues are known for hyper operators. At least, the complex extension of tetration is much more difficult than exponentiation.

## Properties of exponentiation

The following are identities of exponentiation:

- \[a^0 = 1\]
- \[a^1 = a\]
- \[1^a = 1\]
- \[0^a = 0\]

Here \(a\) is a real number, and is non-zero in the fourth equality. Although we set \(0^0 := 1\) in the previous section in order to define \(e^x\), \(0^0\) has different values, e.g. \(0\) or \(1\), depending on context; it is often treated as undefined for this purpose.

The following are some useful properties in manipulating exponents:

- \[a^{-b} = \frac{1}{a^b}\]
- \[a^{b + c} = a^b \cdot a^c\]
- \[a^{b - c} = \frac{a^b}{a^c}\]
- \[a^{b \cdot c} = \left(a^b\right)^c\]

Here variables in each equality are real numbers such that both hand sides make sense. For example, in the first equiality, \(a\) should be non-zero, and should be positive unless \(b\) is an integer.

These can be proved by expressing the exponents in terms of the exponential function.

For a non-negative real number \(a\) and a positive integer \(b\), \(a^{1/b}\) is often written \(\sqrt[b]{a}\), called **radical notation**. When \(b = 2\), it is usually left out: \(\sqrt{a}\). This is called the **square root** of \(a\).

Unlike the previous two hyper-operators, i.e. addition and multiplication, exponentiation is neither commutative nor associative. For example, \(3^5 = 243 > 125 = 5^3\), and \(3^{2^3} = 6,561 > 729 = \left(3^2\right)^3\).

This should be noted that \(a^b\ne b^a\) except when they are both same or they are 2 and 4 when they are integers, though they the hyper operator on the positive integers.

Repeated exponentiation is solved from right to left. For example, a^{bcd} = a^{(b(cd))}.

If the exponentiation is with other operators in the math sentences, the ^ will be solved first like a*b^c = a*(b^c).

## Applications

In calculus

Two important rules of calculus are the **Power Rules** of Differentiation and Integration:

\[\frac{d}{dx}x^n = nx^{n - 1}\]

\[\int x^n dx = \frac{1}{n + 1}x^{n + 1} + C\]

Here \(n\) is a real number, and satisfies \(n \neq -1\) in the second equality. The domain of the functions in the right hand sides depends on the value of \(n\).

### Notations

The exponential function \(a^b\) can be represented:

- In arrow notation as \(a \uparrow b\).
- In chained arrow notation as \(a \rightarrow b\) or \(a \rightarrow b \rightarrow 1\).
- In BEAF as \(\{a, b\}\) or \(a\ \{1\}\ b\).
^{[1]} - In Hyper-E notation as E(a)b.
- In plus notation as \(a +++ b\).
- In star notation (as used in the Big Psi project) as \(a ** b\).
- In the programming languages Python and Ruby, it is written as
`a ** b`

.

## Special exponents

The case \(a^2\) is called the **square** of \(a\), because it is the area of a square with side length \(a\). Likewise, \(a^3\) is the **cube** of \(a\). \(a^4\) is sometimes called the **tesseract** of \(a\), but this term is not used frequently.

## Sources

## See also

- -plex
- Category:Hypercubes from Verse and Dimensions Wiki
- Exponential factorial
- Googology Course

**Hyper-operators:**successor · addition · multiplication ·

**exponentiation**· tetration · pentation · hexation · heptation · octation · enneation · decation (un/doe/tre) · vigintation (doe) · trigintation · centation (tri) · docentation · myriation · circulation

**Bowers' extensions:**expansion · multiexpansion · powerexpansion · expandotetration · explosion (multi/power/tetra) · detonation · pentonation

**Saibian's extensions:**hexonation · heptonation · octonation · ennonation · deconation

**Tiaokhiao's extensions:**megotion (multi/power/tetra) · megoexpansion (multi/power/tetra) · megoexplosion · megodetonation · gigotion (expand/explod/deto) · terotion · more...