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.

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

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

Formally, \(a^b := e^{b \ln a}\), where \(e^x\) is the exponential function and \(\ln\) is the natural logarithm, which are defined like so:

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

This allows the definition to be expanded to non-integer exponents.

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 \(3^{2^3} = 6561 > 729 = \left(3^2\right)^3\).

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

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

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

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.