Exponentiation

 ( Exponentials ) 

          b^m \cdot b^n &= b^{m + n} \\
 \left(b^m\right)^n &= b^{m \cdot n} \\
      b^n \cdot c^n &= (b \cdot c)^n
     

Exponentiation with integer exponents obeys the following laws, for and in the algebraic structure, and and integers:

$\backslash begin\{align\}$

These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrix (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and of any mathematical structure.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if is a real function whose valued can be multiplied, $f^n$ denotes the exponentiation with respect of multiplication, and $f^\{\backslash circ\; n\}$ may denote exponentiation with respect of function composition. That is,

$(f^n)(x)=(f(x))^n=f(x)\; \backslash ,f(x)\; \backslash cdots\; f(x),$

$(f^\{\backslash circ\; n\})(x)=f(f(\backslash cdots\; f(f(x))\backslash cdots)).$

So, if is a group, $x^n$ is defined for every $x\backslash in\; G$ and every integer .

The set of all powers of an element of a group form a subgroup. A group (or subgroup) that consists of all powers of a specific element is the cyclic group generated by . If all the powers of are distinct, the group is isomorphic to the additive group $\backslash Z$ of the integers. Otherwise, the cyclic group is finite group (it has a finite number of elements), and its number of elements is the order of . If the order of is , then $x^n=x^0=1,$ and the cyclic group generated by consists of the first powers of (starting indifferently from the exponent or ).

Order of elements play a fundamental role in group theory. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible orders of group elements are important in the study of the structure of a group (see Sylow theorems), and in the classification of finite simple groups.

Superscript notation is also used for conjugacy class; that is, , where and are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely $(g^h)^k=g^\{hk\}$ and $(gh)^k=g^kh^k.$

If the nilradical is reduced to the zero ideal (that is, if $x\backslash neq\; 0$ implies $x^n\backslash neq\; 0$ for every positive integer ), the commutative ring is said to be reduced ring. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.

More generally, given an ideal in a commutative ring , the set of the elements of that have a power in is an ideal, called the radical of . The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring $kx\_1,$ over a field , an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of Hilbert's Nullstellensatz).

Matrix powers appear often in the context of discrete dynamical systems, where the matrix expresses a transition from a state vector of some system to the next state of the system. This is the standard interpretation of a Markov chain, for example. Then $A^2x$ is the state of the system after two time steps, and so forth: $A^nx$ is the state of the system after time steps. The matrix power $A^n$ is the transition matrix between the state now and the state at a time steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.

Apart from matrices, more general can also be exponentiated. An example is the derivative operator of calculus, $d/dx$, which is a linear operator acting on functions $f(x)$ to give a new function $(d/dx)f(x)\; =\; f\text{'}(x)$. The th power of the differentiation operator is the th derivative:

$\backslash left(\backslash frac\{d\}\{dx\}\backslash right)^nf(x)\; =\; \backslash frac\{d^n\}\{dx^n\}f(x)\; =\; f^\{(n)\}(x).$

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups.E. Hille, R. S. Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975. Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

A finite field is a field with a finite set of elements. This number of elements is either a prime number or a prime power; that is, it has the form $q=p^k,$ where is a prime number, and is a positive integer. For every such , there are fields with elements. The fields with elements are all isomorphic, which allows, in general, working as if there were only one field with elements, denoted $\backslash mathbb\; F\_q.$

One has

$x^q=x$

A primitive element in $\backslash mathbb\; F\_q$ is an element such that the set of the first powers of (that is, $\backslash \{g^1=g,\; g^2,\; \backslash ldots,\; g^\{p-1\}=g^0=1\backslash \}$) equals the set of the nonzero elements of $\backslash mathbb\; F\_q.$ There are $\backslash varphi\; (p-1)$ primitive elements in $\backslash mathbb\; F\_q,$ where $\backslash varphi$ is Euler's totient function.

In $\backslash mathbb\; F\_q,$ the freshman's dream identity

$(x+y)^p\; =\; x^p+y^p$

$\backslash begin\{align\}$

The Diffie–Hellman key exchange is an application of exponentiation in finite fields that is widely used for secure communications. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, if is a primitive element in $\backslash mathbb\; F\_q,$ then $g^e$ can be efficiently computed with exponentiation by squaring for any , even if is large, while there is no known computationally practical algorithm that allows retrieving from $g^e$ if is sufficiently large.

This allows defining the th power $S^n$ of a set as the set of all - $(x\_1,\; \backslash ldots,\; x\_n)$ of elements of .

When is endowed with some structure, it is frequent that $S^n$ is naturally endowed with a similar structure. In this case, the term "direct product" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example $\backslash R^n$ (where $\backslash R$ denotes the real numbers) denotes the Cartesian product of copies of $\backslash R,$ as well as their direct product as vector space, topological spaces, rings, etc.

Given two sets and , the set of all functions from to is denoted $S^T$. This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):

$(S^T)^U\backslash cong\; S^\{T\backslash times\; U\},$

$S^\{T\backslash sqcup\; U\}\backslash cong\; S^T\backslash times\; S^U,$

One can use sets as exponents for other operations on sets, typically for of , , or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, $\backslash R^\backslash N$ denotes the vector space of the infinite sequences of real numbers, and $\backslash R^\{(\backslash N)\}$ the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals , while the Hamel basis of the former cannot be explicitly described (because their existence involves Zorn's lemma).

In this context, can represents the set $\backslash \{0,1\backslash \}.$ So, $2^S$ denotes the power set of , that is the set of the functions from to $\backslash \{0,1\backslash \},$ which can be identified with the set of the of , by mapping each function to the inverse image of .

This fits in with the exponentiation of cardinal numbers, in the sense that , where is the cardinality of .

$\backslash hom(U,S^T)\backslash cong\; \backslash hom(T\backslash times\; U,S).$

This generalizes to the definition of exponentiation in a category in which finite exist: in such a category, the functor $X\backslash to\; X^T$ is, if it exists, a right adjoint to the functor $Y\backslash to\; T\backslash times\; Y.$ A category is called a Cartesian closed category, if direct products exist, and the functor $Y\backslash to\; X\backslash times\; Y$ has a right adjoint for every .

More precisely, consider the function $f(x,y)\; =\; x^y$ defined on $D\; =\; \backslash \{(x,\; y)\; \backslash in\; \backslash mathbf\{R\}^2\; :\; x\; >\; 0\; \backslash \}$. Then can be viewed as a subset of (that is, the set of all pairs with , belonging to the extended real number line , endowed with the product topology), which will contain the points at which the function has a limit.

In fact, has a limit at all accumulation points of , except for , , and .Nicolas Bourbaki, Topologie générale, V.4.2. Accordingly, this allows one to define the powers by continuity whenever , , except for , , and , which remain indeterminate forms.

Under this definition by continuity, we obtain:

• and , when .
• and , when .
• and , when .
• and , when .

These powers are obtained by taking limits of for positive values of . This method does not permit a definition of when , since pairs with are not accumulation points of .

On the other hand, when is an integer, the power is already meaningful for all values of , including negative ones. This may make the definition obtained above for negative problematic when is odd, since in this case as tends to through positive values, but not negative ones.

$100\; =\; 2^2\; +2^5\; +\; 2^6\; =\; 2^2(1+2^3(1+2))$.

In general, the number of multiplication operations required to compute can be reduced to $\backslash sharp\; n\; +\backslash lfloor\; \backslash log\_\{2\}\; n\backslash rfloor\; -1,$ by using exponentiation by squaring, where $\backslash sharp\; n$ denotes the number of s in the binary representation of . For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available. However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

$(g\backslash circ\; f)(x)=g(f(x))$

If the domain of a function equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the th power of the function under composition, commonly called the th iterate of the function. Thus $f^n$ denotes generally the th iterate of ; for example, $f^3(x)$ means $f(f(f(x))).$

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration before the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication after the parentheses. Thus $f^2(x)=\; f(f(x)),$ and $f(x)^2=\; f(x)\backslash cdot\; f(x).$ When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example $f^\{\backslash circ\; 3\}=f\backslash circ\; f\; \backslash circ\; f,$ and $f^3=f\backslash cdot\; f\backslash cdot\; f.$ For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions. So, $\backslash sin^2\; x$ and $\backslash sin^2(x)$ both mean $\backslash sin(x)\backslash cdot\backslash sin(x)$ and not $\backslash sin(\backslash sin(x)),$ which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.

In this context, the exponent $-1$ denotes always the inverse function, if it exists. So $\backslash sin^\{-1\}x=\backslash sin^\{-1\}(x)\; =\; \backslash arcsin\; x.$ For the multiplicative inverse fractions are generally used as in $1/\backslash sin(x)=\backslash frac\; 1\{\backslash sin\; x\}.$

• x ^ y: AWK, BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua, and most computer algebra systems.
• x ** y. The Fortran character set did not include lowercase characters or punctuation symbols other than +-*/()&=.,' and so used ** for exponentiation (the initial version used a xx b instead.). Many other languages followed suit: Ada, Z shell, KornShell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, Apache Groovy, JavaScript, OCaml, Object REXX, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, and VHDL.
• x ↑ y: Algol Reference language, Commodore BASIC, TRS-80 Level II/III BASIC.
• x ^^ y: Haskell (for fractional base, integer exponents), D.
• x⋆y: APL.

In most programming languages with an infix exponentiation operator, it is right-associative, that is, a^b^c is interpreted as a^(b^c).