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.

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
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 and \(\ln\) is the, 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
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 \(a^b \not= b^a\) except when they are both same or they are 1 or 2, 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 sentances, the ^ will be solved first like: a*b^c = a*(b^c).

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
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 the term is rare.