**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**.

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

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

## Definition Edit

For nonnegative integers \(b\), exponentiation has the following definition:

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

For general values of \(b\), \(a^b\) 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:

\[e^x := 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\] \[\ln x := \int_1^x \frac{dt}{t}\] (where 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.

## Properties of exponentiation Edit

The following are identities of exponentiation:

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

\(0^0 = 0 \text{ or } 1\) has different values depending on context; it is often treated as undefined for this purpose.

The following are some useful properties in manipulating exponents:

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

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

\(a^{1/b}\) 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, 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 sould be noted that \(a^b\ne b^a\) except when they are both same or they are 2 and 4 when they are integer, 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 sentances, the ^ will be solved first like: a*b^c = a*(b^c) .

## ApplicationsEdit

In calculus

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

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

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

### Notations Edit

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 Edit

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.

## SourcesEdit

## See also Edit

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

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

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