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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 Multiplication, It is: .

In Addition, It is: .

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 non-zero real number \(a\) and a negative integer \(b\), the exponentiation has the following definition using the non-negative exponent \(-b\).

\[a^b := \frac{1}{a^{-b}}\]

For a positive real number \(a\) and a rational number \(b\), the exponentiation has the following definition using the integer exponent \(n\):

\[a^b := \textrm{the } m \textrm{-th root of } a^n\]

(When there are two \(m\)-th roots, choose the positive one.) Here \(m\) is a positive integer and \(n\) is an integer such that \(mb = n\). The existence of an \(m\)-th root follows from the continuity of real numbers (especially intermediate value theorem). Althouh the choice of such an \((m,n)\) is not unique, \(a^b\) is known to be well-defined by exponential laws in #Properties of exponentiation for integer variables.

For a positive real number \(a\) and a real number \(b\), the exponentiation has the following definition using the rational exponents \(b_n\):

\[a^b := \lim_{n \to \infty} a^{b_n}\]

Here \((b_n)_{n \in \mathbb{N}}\) is a rational sequences converging to \(b\). Although the choice of such a \((b_n)_{n \in \mathbb{N}}\) is not unique, \(a^b\) is known to be well-defined by the exponential laws and the continuity of the exponentiation (with rational \(b\) in order not to cause circular logic) at \(b = 0\).

In the same setting (or a partially wider setting like the case for complex numbers as we will explain later), the exponentiation can be defined also in the following way:

\[a^b := e^{b \ln a}\]

Here \(e^x\) is the exponential function and \(\ln\) is the natural logarithm, 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-rational 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 induction when the variables run only though natural number. For the case where the variables run through integers, these can be proved by the results for the case where the variables run through natural numbers. Similarly, these for a generalised case can be proved by the results for a less general case. These can also be proved by expressing the exponents in terms of the exponential function, as long as the coincidence of the formulations above, which requires a similar argument at last, is proved.

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 \(3^{2^3} = 6,561 > 729 = \left(3^2\right)^3\). Note that when \(a\) and \(b\) are integers, \(a^b\ne b^a\) except when \(a=b\) or they are 2 and 4.

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 \(a\ \{1\}\ b\).[1]
  • 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 Python and Ruby, 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.

Sources

See also

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