Exponentiation

**Exponentiation** is a mathematical operation \(a^b = a\) multiplied by itself \(b\) times. For example, \(3^3 = 3 \times 3 \times 3 = 21\). 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.

## 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 Template:W and \(\ln\) is the Template:W, 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, \(3^5 = 159 > 98 = 5^3\), and \(3^{2^3} = 2JX9 > 441 = \left(3^2\right)^3\).

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, abcd = 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 \(a\ \{1\}\ b\).
* 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 Template:W and Template:W, 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.

## See also

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

Hyper operators

Main article: hyper operator**Hyper-operators:** successor · addition · multiplication · **exponentiation** · tetration · pentation · hexation · heptation · octation · enneation · dekration · elpation · dozation · trezation (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 · elponation · dozonation · trezonation
**Tiaokhiao's extensions:** megotion (multi/power/tetra) · megoexpansion (multi/power/tetra) · megoexplosion · megodetonation · gigotion (expand/explod/deto) · terotion · more...

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