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. \(a^b\) is equivalent to fill b-dimensional with sidelength a. It can also be expressed as {a,b} in BEAF, or a⋅⋅b in programming languages such as Python or Ruby. The repeated exponention is called tetration.

It is the third hyper operator.

In the fast-growing hierarchy, \(f_2(n)\) corresponds to exponantiational 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:

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

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

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.

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.