Exponentiation

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.

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 and \(\ln\) is the, 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, ab c d = a(b (c d) ).

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 and, it is written as.

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.