{"history":[],"article":{"id":"51f1fbcd-a4c9-44c8-abc9-9ceab1abeaf3","title":"Exponentiation","slug":"exponentiation","content":":::summary\n**Mark:** bn \n**Name:** notation \n**Variant 1 Caption:** base b and exponent n \n:::\n\n# \n[[File:Expo02.svg|thumb|315px|Graphs of\ny = bx\nfor various bases\nb\nEach curve passes through the point\n(0, 1)\nbecause any nonzero number raised to the power of\n0\nis\n1\n. At\nx = 1\n, the value of\ny\nequals the base because any number raised to the power of\n1\nis the number itself.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), **exponentiation** is an [operation](https://bliptext.com/articles/operation-mathematics) involving [two numbers](https://bliptext.com/articles/binary-operation): the [base](https://bliptext.com/articles/base-exponentiation) and the exponent or power.\nExponentiation is written as\nbn\n, where b is the base and n is the power; often said as \"b to the power n\".\nWhen n is a positive [integer](https://bliptext.com/articles/integer), exponentiation corresponds to repeated [multiplication](https://bliptext.com/articles/multiplication) of the base: that is,\nbn\nis the [product](https://bliptext.com/articles/product-mathematics) of multiplying n bases:\nIn particular, b^1=b.\nThe exponent is usually shown as a [superscript](https://bliptext.com/articles/superscript) to the right of the base as\nbn\nor in computer code as b^n, and may also be called \"b raised to the nth power\", \"b to the power of n\", \"the nth power of b\", or most briefly \"b to the n\".\nThe above definition of b^n immediately implies several properties, in particular the multiplication rule:\nThat is, when multiplying a base raised to one power times the same base raised to another power, the powers add.\nExtending this rule to the power zero gives, and dividing both sides by b^n gives.\nThat is, the multiplication rule implies the definition b^0=1.A similar argument implies the definition for negative integer powers: That is, extending the multiplication rule gives.\nDividing both sides by b^n gives.\nThis also implies the definition for fractional powers: For example,, meaning , which is the definition of square root:.\nThe definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define b^x for any positive real base b and any real number exponent x. More involved definitions allow [complex](https://bliptext.com/articles/complex-numbers) base and exponent, as well as certain types of [matrices](https://bliptext.com/articles/matrix-mathematics) as base or exponent.\nExponentiation is used extensively in many fields, including [economics](https://bliptext.com/articles/economics), [biology](https://bliptext.com/articles/biology), [chemistry](https://bliptext.com/articles/chemistry), [physics](https://bliptext.com/articles/physics), and [computer science](https://bliptext.com/articles/computer-science), with applications such as [compound interest](https://bliptext.com/articles/compound-interest), [population growth](https://bliptext.com/articles/population-growth), [chemical reaction kinetics](https://bliptext.com/articles/chemical-reaction-kinetics), [wave](https://bliptext.com/articles/wave) behavior, and [public-key cryptography](https://bliptext.com/articles/public-key-cryptography).\n\n# Etymology\nThe term exponent originates from the [Latin](https://bliptext.com/articles/latin) exponentem, the [present participle](https://bliptext.com/articles/present-participle) of exponere, meaning \"to put forth\".\nThe term power is a mistranslation of the [ancient Greek](https://bliptext.com/articles/ancient-greek) δύναμις (dúnamis, here: \"amplification\" ) used by the [Greek](https://bliptext.com/articles/greek-mathematics) mathematician [Euclid](https://bliptext.com/articles/euclid) for the square of a line, following [Hippocrates of Chios](https://bliptext.com/articles/hippocrates-of-chios).\n\n# History\n\n## Antiquity\n\n### The Sand Reckoner\nIn [The Sand Reckoner](https://bliptext.com/articles/the-sand-reckoner), [Archimedes](https://bliptext.com/articles/archimedes) proved the law of exponents,\n10a · 10b = 10a+b\n, necessary to manipulate powers of\n10\n. He then used powers of\n10\nto estimate the number of grains of sand that can be contained in the universe.\n\n## Islamic Golden Age\n\n### Māl and kaʿbah (\"square\" and \"cube\")\nIn the 9th century, the Persian mathematician [Al-Khwarizmi](https://bliptext.com/articles/al-khwarizmi) used the terms مَال (māl, \"possessions\", \"property\") for a [square](https://bliptext.com/articles/square-algebra)—the Muslims, \"like most mathematicians of those and earlier times, thought of a [square](https://bliptext.com/articles/square-algebra)d number as a depiction of an area, especially of land, hence property\" —and كَعْبَة ([Kaʿbah](https://bliptext.com/articles/kaaba), \"[cube](https://bliptext.com/articles/cube-algebra)\") for a [cube](https://bliptext.com/articles/cube-algebra), which later [Islamic](https://bliptext.com/articles/mathematics-in-the-medieval-islamic-world) mathematicians represented in [mathematical notation](https://bliptext.com/articles/mathematical-notation) as the letters [mīm](https://bliptext.com/articles/m-m) (m) and [kāf](https://bliptext.com/articles/k-f) (k), respectively, by the 15th century, as seen in the work of [Abu'l-Hasan ibn Ali al-Qalasadi](https://bliptext.com/articles/abu-l-hasan-ibn-ali-al-qalasadi).\n\n## 15th–18th century\n\n### Introducing exponents\n[Nicolas Chuquet](https://bliptext.com/articles/nicolas-chuquet) used a form of exponential notation in the 15th century, for example\n122\nto represent\n12x2\n. This was later used by [Henricus Grammateus](https://bliptext.com/articles/henricus-grammateus) and [Michael Stifel](https://bliptext.com/articles/michael-stifel) in the 16th century.\nIn the late 16th century, [Jost Bürgi](https://bliptext.com/articles/jost-b-rgi) would use Roman numerals for exponents in a way similar to that of Chuquet, for example iii 4 for\n4x3\n.\n\n### \"Exponent\"; \"square\" and \"cube\"\nThe word exponent was coined in 1544 by Michael Stifel.\nIn the 16th century, [Robert Recorde](https://bliptext.com/articles/robert-recorde) used the terms square, cube, zenzizenzic ([fourth power](https://bliptext.com/articles/fourth-power)), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and [zenzizenzizenzic](https://bliptext.com/articles/zenzizenzizenzic) (eighth).\nBiquadrate has been used to refer to the fourth power as well.\n\n### Modern exponential notation\nIn 1636, [James Hume](https://bliptext.com/articles/james-hume-mathematician) used in essence modern notation, when in L'algèbre de Viète he wrote\nAiii\nfor\nA3\n. Early in the 17th century, the first form of our modern exponential notation was introduced by [René Descartes](https://bliptext.com/articles/ren-descartes) in his text titled [La Géométrie](https://bliptext.com/articles/la-g-om-trie); there, the notation is introduced in Book I.\n\"I designate ...\naa\n, or\na2\nin multiplying\na\nby itself; and\na3\nin multiplying it once more again by\na\n, and thus to infinity.\"\nSome mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication.\nThus they would write [polynomials](https://bliptext.com/articles/polynomial), for example, as\nax + bxx + cx3 + d\n.\n\n### \"Indices\"\n[Samuel Jeake](https://bliptext.com/articles/samuel-jeake) introduced the term indices in 1696.\nThe term involution was used synonymously with the term indices, but had declined in usage and should not be confused with [its more common meaning](https://bliptext.com/articles/involution-mathematics).\n\n### Variable exponents, non-integer exponents\nIn 1748, [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) introduced variable exponents, and, implicitly, non-integer exponents by writing:\"Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant.\"\n\n## 20th century\nAs calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation.\nFor example [Konrad Zuse](https://bliptext.com/articles/konrad-zuse) introduced [floating point arithmetic](https://bliptext.com/articles/floating-point-arithmetic) in his 1938 computer Z1.\nOne [register](https://bliptext.com/articles/register-computer) contained representation of leading digits, and a second contained representation of the exponent of 10.\nEarlier [Leonardo Torres Quevedo](https://bliptext.com/articles/leonardo-torres-quevedo) contributed Essays on Automation (1914) which had suggested the floating-point representation of numbers.\nThe more flexible [decimal floating-point](https://bliptext.com/articles/decimal-floating-point) representation was introduced in 1946 with a [Bell Laboratories](https://bliptext.com/articles/bell-laboratories) computer.\nEventually educators and engineers adopted [scientific notation](https://bliptext.com/articles/scientific-notation) of numbers, consistent with common reference to [order of magnitude](https://bliptext.com/articles/order-of-magnitude) in a [ratio scale](https://bliptext.com/articles/ratio-scale).\n\n# Terminology\nThe expression\nb2 = b · b\nis called \"the [square](https://bliptext.com/articles/square-algebra) of\nb\n\" or \"\nb\nsquared\", because the area of a square with side-length\nb\nis\nb2\n. (It is true that it could also be called \"\nb\nto the second power\", but \"the square of\nb\n\" and \"\nb\nsquared\" are more traditional)\nSimilarly, the expression\nb3 = b · b · b\nis called \"the [cube](https://bliptext.com/articles/cube-algebra) of\nb\n\" or \"\nb\ncubed\", because the volume of a cube with side-length\nb\nis\nb3\n.\nWhen an exponent is a [positive integer](https://bliptext.com/articles/positive-integer), that exponent indicates how many copies of the base are multiplied together.\nFor example,\n35 = 3 · 3 · 3 · 3 · 3 = 243\n. The base\n3\nappears\n5\ntimes in the multiplication, because the exponent is\n5\n. Here,\n243\nis the 5th power of 3, or 3 raised to the 5th power.\nThe word \"raised\" is usually omitted, and sometimes \"power\" as well, so\n35\ncan be simply read \"3 to the 5th\", or \"3 to the 5\".\n\n# Integer exponents\nThe exponentiation operation with integer exponents may be defined directly from elementary [arithmetic operations](https://bliptext.com/articles/arithmetic-operation).\n\n## Positive exponents\nThe definition of the exponentiation as an iterated multiplication can be [formalized](https://bliptext.com/articles/formal-proof) by using [induction](https://bliptext.com/articles/mathematical-induction), and this definition can be used as soon as one has an [associative](https://bliptext.com/articles/associativity) multiplication:\nThe base case is and the [recurrence](https://bliptext.com/articles/recurrence-relation) is\nThe associativity of multiplication implies that for any positive integers m and n, and\n\n## Zero exponent\nAs mentioned earlier, a (nonzero) number raised to the\n0\npower is\n1\nThis value is also obtained by the [empty product](https://bliptext.com/articles/empty-product) convention, which may be used in every [algebraic structure](https://bliptext.com/articles/algebraic-structure) with a multiplication that has an [identity](https://bliptext.com/articles/multiplicative-identity).\nThis way the formula\nalso holds for n=0.\nThe case of\n00\nis controversial.\nIn contexts where only integer powers are considered, the value\n1\nis generally assigned to\n00\nbut, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.\n\n## Negative exponents\nExponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b: Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (\\infty).\nThis definition of exponentiation with negative exponents is the only one that allows extending the identity to negative exponents (consider the case m=-n).\nThe same definition applies to [invertible elements](https://bliptext.com/articles/invertible-element) in a multiplicative [monoid](https://bliptext.com/articles/monoid), that is, an [algebraic structure](https://bliptext.com/articles/algebraic-structure), with an associative multiplication and a [multiplicative identity](https://bliptext.com/articles/multiplicative-identity) denoted\n1\n(for example, the [square matrices](https://bliptext.com/articles/square-matrix) of a given dimension).\nIn particular, in such a structure, the inverse of an [invertible element](https://bliptext.com/articles/invertible-element) x is standardly denoted x^{-1}.\n\n## Identities and properties\nThe following [identities](https://bliptext.com/articles/identity-mathematics), often called **exponent rules**, hold for all integer exponents, provided that the base is non-zero:\nUnlike addition and multiplication, exponentiation is not [commutative](https://bliptext.com/articles/commutative): for example, 2^3 = 8, but reversing the operands gives the different value 3^2=9.\nAlso unlike addition and multiplication, exponentiation is not [associative](https://bliptext.com/articles/associative): for example,\n(23)2 = 82 = 64\n, whereas\n2(3 2) = 29 = 512\n. Without parentheses, the conventional [order of operations](https://bliptext.com/articles/order-of-operations) for [serial exponentiation](https://bliptext.com/articles/serial-exponentiation) in superscript notation is top-down (or right-associative), not bottom-up (or left-associative).\nThat is,\nwhich, in general, is different from\n\n## Powers of a sum\nThe powers of a sum can normally be computed from the powers of the summands by the [binomial formula](https://bliptext.com/articles/binomial-formula)\nHowever, this formula is true only if the summands commute (i.e. that\nab = ba\n), which is implied if they belong to a [structure](https://bliptext.com/articles/algebraic-structure) that is [commutative](https://bliptext.com/articles/commutative-property). Otherwise, if a and b are, say, [square matrices](https://bliptext.com/articles/square-matrix) of the same size, this formula cannot be used. It follows that in [computer algebra](https://bliptext.com/articles/computer-algebra), many [algorithms](https://bliptext.com/articles/algorithm) involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose [computer algebra](https://bliptext.com/articles/computer-algebra) systems use a different notation (sometimes\n^^\ninstead of\n^\n) for exponentiation with non-commuting bases, which is then called **non-commutative exponentiation**.\n\n## Combinatorial interpretation\nFor nonnegative integers n and m, the value of\nnm\nis the number of [functions](https://bliptext.com/articles/function-mathematics) from a [set](https://bliptext.com/articles/set-mathematics) of m elements to a [set](https://bliptext.com/articles/set-mathematics) of n elements (see [cardinal exponentiation](https://bliptext.com/articles/cardinal-exponentiation)).\nSuch functions can be represented as m-[tuples](https://bliptext.com/articles/tuple) from an n-element set (or as m-letter words from an n-letter alphabet).\nSome examples for particular values of m and n are given in the following table:\n\n## Particular bases\n\n### Powers of ten\nIn the base ten ([decimal](https://bliptext.com/articles/decimal)) number system, integer powers of\n10\nare written as the digit\n1\nfollowed or preceded by a number of zeroes determined by the sign and magnitude of the exponent.\nFor example,\n= 1,000\nand\n= 0\n.\nExponentiation with base\n10\nis used in [scientific notation](https://bliptext.com/articles/scientific-notation) to denote large or small numbers.\nFor instance, 299,792,458 m/s (the [speed of light](https://bliptext.com/articles/speed-of-light) in vacuum, in [metres per second](https://bliptext.com/articles/metres-per-second)) can be written as 2.998 m/s and then [approximated](https://bliptext.com/articles/approximation) as 2.998 m/s.\n[SI prefixes](https://bliptext.com/articles/si-prefix) based on powers of\n10\nare also used to describe small or large quantities.\nFor example, the prefix [kilo](https://bliptext.com/articles/kilo) means\n= 1,000\n, so a kilometre is 1,000 metres.\n\n### Powers of two\nThe first negative powers of\n2\nhave special names: 2^{-1}is a [half](https://bliptext.com/articles/one-half); 2^{-2} is a [quarter](https://bliptext.com/articles/4-number).\nPowers of\n2\nappear in [set theory](https://bliptext.com/articles/set-theory), since a set with\nn\nmembers has a [power set](https://bliptext.com/articles/power-set), the set of all of its [subsets](https://bliptext.com/articles/subset), which has\n2n\nmembers.\nInteger powers of\n2\nare important in [computer science](https://bliptext.com/articles/computer-science).\nThe positive integer powers\n2n\ngive the number of possible values for an\nn\n-[bit](https://bliptext.com/articles/bit) integer [binary number](https://bliptext.com/articles/binary-number); for example, a [byte](https://bliptext.com/articles/byte) may take\n28 = 256\ndifferent values.\nThe [binary number system](https://bliptext.com/articles/binary-number-system) expresses any number as a sum of powers of\n2\n, and denotes it as a sequence of\n0\nand\n1\n, separated by a [binary point](https://bliptext.com/articles/binary-point), where\n1\nindicates a power of\n2\nthat appears in the sum; the exponent is determined by the place of this\n1\n1\non the left of the point (starting from\n0\n), and the negative exponents are determined by the rank on the right of the point.\n\n### Powers of one\nEvery power of one equals:\n1n = 1\n.\n\n### Powers of zero\nFor a positive exponent\nn > 0\n, the nth power of zero is zero:\n0n = 0\n. For a negative\\ exponent, is undefined.\nThe expression [[zero to the power of zero|\n00\n]] is either [defined as](https://bliptext.com/articles/indeterminate-form), or it is left undefined.\n\n### Powers of negative one\nSince a negative number times another negative is positive, we have: Because of this, powers of\n−1\nare useful for expressing alternating [sequences](https://bliptext.com/articles/sequence).\nFor a similar discussion of powers of the complex number\ni\n, see .\n\n## Large exponents\nThe [limit of a sequence](https://bliptext.com/articles/limit-of-a-sequence) of powers of a number greater than one diverges; in other words, the sequence grows without bound:\nbn → ∞\nas\nn → ∞\nwhen\nb > 1\nThis can be read as \"b to the power of n tends to [+∞](https://bliptext.com/articles/extended-real-number-line) as n tends to infinity when b is greater than one\".\nPowers of a number with [absolute value](https://bliptext.com/articles/absolute-value) less than one tend to zero:\nbn → 0\nas\nn → ∞\nwhen\nAny power of one is always one:\nbn = 1\nfor all\nn\nfor\nb = 1\nPowers of a negative number b\\leq -1 alternate between positive and negative as\nn\nalternates between even and odd, and thus do not tend to any limit as\nn\ngrows.\nIf the exponentiated number varies while tending to\n1\nas the exponent tends to infinity, then the limit is not necessarily one of those above.\nA particularly important case is\n(1 + 1/n)n → e\nas\nn → ∞\nSee below.\nOther limits, in particular those of expressions that take on an [indeterminate form](https://bliptext.com/articles/indeterminate-form), are described in below.\n\n## Power functions\n[[File:Potenssi 1 3 5.svg|thumb|left|Power functions for\nn = 1, 3, 5\n]] [[File:Potenssi 2 4 6.svg|thumb|Power functions for\nn = 2, 4, 6\n]]\nReal functions of the form f(x) = cx^n, where c \\ne 0, are sometimes called power functions.\nWhen n is an [integer](https://bliptext.com/articles/integer) and n \\ge 1, two primary families exist: for n even, and for n odd.\nIn general for c > 0, when n is even f(x) = cx^n will tend towards positive [infinity](https://bliptext.com/articles/infinity-mathematics) with increasing x, and also towards positive [infinity](https://bliptext.com/articles/infinity-mathematics) with decreasing x. All graphs from the family of even power functions have the general shape of y=cx^2, flattening more in the middle as n increases.\nFunctions with this kind of [symmetry](https://bliptext.com/articles/symmetry) are called [even functions](https://bliptext.com/articles/even-functions).\nWhen n is odd, f(x)'s [asymptotic](https://bliptext.com/articles/asymptotic) behavior reverses from positive x to negative x. For c > 0, f(x) = cx^n will also tend towards positive [infinity](https://bliptext.com/articles/infinity-mathematics) with increasing x, but towards negative [infinity](https://bliptext.com/articles/infinity-mathematics) with decreasing x. All graphs from the family of odd power functions have the general shape of y=cx^3, flattening more in the middle as n increases and losing all flatness there in the straight line for n=1.\nFunctions with this kind of symmetry are called [odd functions](https://bliptext.com/articles/odd-function).\nFor c < 0, the opposite asymptotic behavior is true in each case.\n\n## Table of powers of decimal digits\n\n# Rational exponents\n[[File:Mplwp roots 01.svg|right|thumb|300px|From top to bottom:\nx1/8\n,\nx1/4\n,\nx1/2\n,\nx1\n,\nx2\n,\nx4\n,\nx8\n.]]\nIf x is a nonnegative [real number](https://bliptext.com/articles/real-number), and n is a positive integer, x^{1/n} or \\sqrt[n]x denotes the unique nonnegative real [nth root](https://bliptext.com/articles/nth-root) of x, that is, the unique nonnegative [real number](https://bliptext.com/articles/real-number) y such that y^n=x.\nIf x is a positive real number, and \\frac pq is a [rational number](https://bliptext.com/articles/rational-number), with p and q > 0 integers, then x^{p/q} is defined as The equality on the right may be derived by setting and writing\nIf r is a positive rational number,\n0r = 0\n, by definition.\nAll these definitions are required for extending the identity to rational exponents.\nOn the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers.\nFor example, a negative real number has a real nth root, which is negative, if n is [odd](https://bliptext.com/articles/odd-number), and no real root if n is even.\nIn the latter case, whichever complex nth root one chooses for x^\\frac 1n, the identity cannot be satisfied.\nFor example,\nSee **' and '** for details on the way these problems may be handled.\n\n# Real exponents\nFor positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (**', below), or in terms of the [logarithm](https://bliptext.com/articles/logarithm) of the base and the [exponential function](https://bliptext.com/articles/exponential-function) ('**, below).\nThe result is always a positive real number, and the [identities and properties](https://bliptext.com/articles/) shown above for integer exponents remain true with these definitions for real exponents.\nThe second definition is more commonly used, since it generalizes straightforwardly to [complex](https://bliptext.com/articles/complex-number) exponents.\nOn the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values.\nOne may choose one of these values, called the [principal value](https://bliptext.com/articles/principal-value), but there is no choice of the [principal value](https://bliptext.com/articles/principal-value) for which the identity\nis true; see .\nTherefore, exponentiation with a basis that is not a positive real number is generally viewed as a [multivalued function](https://bliptext.com/articles/multivalued-function).\n\n## Limits of rational exponents\n[[File:Continuity of the Exponential at 0.svg|thumb|The limit of\ne1/n\nis\ne0 = 1\nwhen n tends to the infinity.]]\nSince any [irrational number](https://bliptext.com/articles/irrational-number) can be expressed as the [limit of a sequence](https://bliptext.com/articles/limit-of-a-sequence) of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by [continuity](https://bliptext.com/articles/continuous-function) with the rule\nwhere the limit is taken over rational values of r only.\nThis limit exists for every positive b and every real x.\nFor example, if\nx = π\n, the [non-terminating decimal](https://bliptext.com/articles/non-terminating-decimal) representation\nπ = 3.14159...\nand the [monotonicity](https://bliptext.com/articles/monotone-function) of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain b^\\pi:\nSo, the upper bounds and the lower bounds of the intervals form two [sequences](https://bliptext.com/articles/sequence-mathematics) that have the same limit, denoted b^\\pi.\nThis defines b^x for every positive b and real x as a [continuous function](https://bliptext.com/articles/continuous-function) of b and x. See also [Well-defined expression](https://bliptext.com/articles/well-defined-expression).\n\n## Exponential function\nThe exponential function may be defined as where is [Euler's number](https://bliptext.com/articles/euler-s-number), but to avoid [circular reasoning](https://bliptext.com/articles/circular-reasoning), this definition cannot be used here.\nRather, we give an independent definition of the exponential function \\exp(x), and of e=\\exp(1), relying only on positive integer powers (repeated multiplication).\nThen we sketch the proof that this agrees with the previous defintion:\nThere are [many equivalent ways to define the exponential function](https://bliptext.com/articles/characterizations-of-the-exponential-function), one of them being\nOne has \\exp(0)=1, and the exponential identity (or multiplication rule) holds as well, since\nand the second-order term does not affect the limit, yielding.\nEuler's number can be defined as e=\\exp(1).\nIt follows from the preceding equations that \\exp(x)=e^x when x is an integer (this results from the repeated-multiplication definition of the exponentiation).\nIf x is real, \\exp(x)=e^x results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity of the exponential function otherwise.\nThe limit that defines the exponential function converges for every [complex](https://bliptext.com/articles/complex-number) value of x, and therefore it can be used to extend the definition of \\exp(z), and thus e^z, from the real numbers to any [complex](https://bliptext.com/articles/complex-number) argument z. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for [complex](https://bliptext.com/articles/complex-number) base and exponent.\n\n## Powers via logarithms\nThe definition of\nex\nas the exponential function allows defining\nbx\nfor every positive real numbers b, in terms of exponential and [logarithm](https://bliptext.com/articles/logarithm) function.\nSpecifically, the fact that the [natural logarithm](https://bliptext.com/articles/natural-logarithm)\nln(x)\nis the [inverse](https://bliptext.com/articles/inverse-function) of the exponential function\nex\nmeans that one has\nfor every\nb > 0\n. For preserving the identity one must have\nSo, e^{x \\ln b} can be used as an alternative definition of\nbx\nfor any positive real b. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.\n\n# Complex exponents with a positive real base\nIf b is a positive real number, exponentiation with base b and [complex](https://bliptext.com/articles/complex-number) exponent z is defined by means of the exponential function with [complex](https://bliptext.com/articles/complex-number) argument (see the end of , above) as where \\ln b denotes the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of b.\nThis satisfies the identity\nIn general,\nis not defined, since\nbz\nis not a real number.\nIf a meaning is given to the exponentiation of a complex number (see , below), one has, in general,\nunless z is real or t is an integer.\n[Euler's formula](https://bliptext.com/articles/euler-s-formula),\nallows expressing the [polar form](https://bliptext.com/articles/polar-form) of b^z in terms of the [real and imaginary parts](https://bliptext.com/articles/real-and-imaginary-parts) of z, namely\nwhere the [absolute value](https://bliptext.com/articles/absolute-value) of the [trigonometric](https://bliptext.com/articles/trigonometry) factor is one.\nThis results from\n\n# Non-integer powers of complex numbers\nIn the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only.\nFor other bases, difficulties appear already with the apparently simple case of nth roots, that is, of exponents 1/n, where n is a positive integer.\nAlthough the general theory of exponentiation with non-integer exponents applies to nth roots, this case deserves to be considered first, since it does not need to use [complex logarithms](https://bliptext.com/articles/complex-logarithm), and is therefore easier to understand.\n\n## nth roots of a complex number\nEvery nonzero complex number z may be written in [polar form](https://bliptext.com/articles/polar-form) as where \\rho is the [absolute value](https://bliptext.com/articles/absolute-value) of z, and \\theta is its [argument](https://bliptext.com/articles/argument-complex-analysis).\nThe argument is defined [up to](https://bliptext.com/articles/up-to) an integer multiple of\n2π\nThe polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments.\nIt follows that the polar form of an nth root of a complex number can be obtained by taking the nth root of the absolute value and dividing its argument by n:\nIf 2\\pi is added to \\theta, the complex number is not changed, but this adds 2i\\pi/n to the argument of the nth root, and provides a new nth root.\nThis can be done n times, and provides the n nth roots of the complex number.\nIt is usual to choose one of the n nth root as the [principal root](https://bliptext.com/articles/principal-root).\nThe common choice is to choose the nth root for which that is, the nth root that has the largest real part, and, if there are two, the one with positive imaginary part.\nThis makes the principal nth root a [continuous function](https://bliptext.com/articles/continuous-function) in the whole complex plane, except for negative real values of the [radicand](https://bliptext.com/articles/radicand).\nThis function equals the usual nth root for positive real radicands.\nFor negative real radicands, and odd exponents, the principal nth root is not real, although the usual nth root is real.\n[Analytic continuation](https://bliptext.com/articles/analytic-continuation) shows that the principal nth root is the unique [complex differentiable](https://bliptext.com/articles/complex-differentiable) function that extends the usual nth root to the complex plane without the nonpositive real numbers.\nIf the complex number is moved around zero by increasing its argument, after an increment of 2\\pi, the complex number comes back to its initial position, and its nth roots are [permuted circularly](https://bliptext.com/articles/circular-permutation) (they are multiplied by e^{2i\\pi/n}).\nThis shows that it is not possible to define a nth root function that is continuous in the whole complex plane.\n\n### Roots of unity\n[[File:One3Root.svg|thumb|right|The three third roots of\n1\n]] The nth roots of unity are the n complex numbers such that\nwn = 1\n, where n is a positive integer.\nThey arise in various areas of mathematics, such as in [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) or algebraic solutions of algebraic equations ([Lagrange resolvent](https://bliptext.com/articles/lagrange-resolvent)).\nThe n nth roots of unity are the n first powers of, that is The nth roots of unity that have this generating property are called primitive nth roots of unity; they have the form with k [coprime](https://bliptext.com/articles/coprime-integers) with n. The unique primitive square root of unity is -1; the primitive fourth roots of unity are i and -i.\nThe nth roots of unity allow expressing all nth roots of a complex number z as the n products of a given nth roots of z with a nth root of unity.\nGeometrically, the nth roots of unity lie on the [unit circle](https://bliptext.com/articles/unit-circle) of the [complex plane](https://bliptext.com/articles/complex-plane) at the vertices of a [regular n-gon](https://bliptext.com/articles/regular-polygon) with one vertex on the real number 1.\nAs the number is the primitive nth root of unity with the smallest positive [argument](https://bliptext.com/articles/argument-complex-analysis), it is called the principal primitive nth root of unity, sometimes shortened as principal nth root of unity, although this terminology can be confused with the [principal value](https://bliptext.com/articles/principal-value) of 1^{1/n}, which is 1.\n\n## Complex exponentiation\nDefining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for z^w.\nSo, either a [principal value](https://bliptext.com/articles/principal-value) is defined, which is not continuous for the values of z that are real and nonpositive, or z^w is defined as a [multivalued function](https://bliptext.com/articles/multivalued-function).\nIn all cases, the [complex logarithm](https://bliptext.com/articles/complex-logarithm) is used to define complex exponentiation as where \\log z is the variant of the [complex logarithm](https://bliptext.com/articles/complex-logarithm) that is used, which is a function or a [multivalued function](https://bliptext.com/articles/multivalued-function) such that\nfor every z in its [domain of definition](https://bliptext.com/articles/domain-of-a-function).\n\n### Principal value\nThe [principal value](https://bliptext.com/articles/principal-value) of the [complex logarithm](https://bliptext.com/articles/complex-logarithm) is the unique continuous function, commonly denoted \\log, such that, for every nonzero complex number z, and the [argument](https://bliptext.com/articles/argument-complex-analysis) of z satisfies\nThe principal value of the complex logarithm is not defined for z=0, it is [discontinuous](https://bliptext.com/articles/continuous-function) at negative real values of z, and it is [holomorphic](https://bliptext.com/articles/holomorphic) (that is, complex differentiable) elsewhere.\nIf z is real and positive, the principal value of the complex logarithm is the natural logarithm:\nThe principal value of z^w is defined as\nwhere \\log z is the principal value of the logarithm.\nThe function is holomorphic except in the neighbourhood of the points where z is real and nonpositive.\nIf z is real and positive, the principal value of z^w equals its usual value defined above.\nIf w=1/n, where n is an integer, this principal value is the same as the one defined above.\n\n### Multivalued function\nIn some contexts, there is a problem with the discontinuity of the principal values of \\log z and z^w at the negative real values of z. In this case, it is useful to consider these functions as [multivalued functions](https://bliptext.com/articles/multivalued-function).\nIf \\log z denotes one of the values of the multivalued logarithm (typically its principal value), the other values are where k is any integer.\nSimilarly, if z^w is one value of the exponentiation, then the other values are given by where k is any integer.\nDifferent values of k give different values of z^w unless w is a [rational number](https://bliptext.com/articles/rational-number), that is, there is an integer d such that dw is an integer.\nThis results from the [periodicity](https://bliptext.com/articles/periodic-function) of the exponential function, more specifically, that e^a=e^b if and only if a-b is an integer multiple of 2\\pi i.\nIf w=\\frac mn is a rational number with m and n [coprime integers](https://bliptext.com/articles/coprime-integers) with n>0, then z^w has exactly n values.\nIn the case m=1, these values are the same as those described in [§ nth roots of a complex number](https://bliptext.com/articles/).\nIf w is an integer, there is only one value that agrees with that of.\nThe multivalued exponentiation is holomorphic for z\\ne 0, in the sense that its [graph](https://bliptext.com/articles/graph-of-a-function) consists of several sheets that define each a holomorphic function in the neighborhood of every point.\nIf z varies continuously along a circle around\n0\n, then, after a turn, the value of z^w has changed of sheet.\n\n### Computation\nThe canonical form x+iy of z^w can be computed from the canonical form of z and w. Although this can be described by a single formula, it is clearer to split the computation in several steps.\n\n#### Examples\n−2\nis So, the above described method gives the values In this case, all the values have the same argument 4\\ln 2, and different absolute values.\nIn both examples, all values of z^w have the same argument.\nMore generally, this is true if and only if the [real part](https://bliptext.com/articles/real-part) of w is an integer.\n\n### Failure of power and logarithm identities\nSome identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions.\nFor example:\n• The identity\nlog(bx) = x ⋅ log b\nholds whenever b is a positive real number and x is a real number.\nBut for the principal branch of the complex logarithm one has\nRegardless of which branch of the logarithm is used, a similar failure of the identity will exist.\nThe best that can be said (if only using this result) is that:\nThis identity does not hold even when considering log as a multivalued function.\nThe possible values of\nlog(wz)\ncontain those of\nz ⋅ log w\nas a proper subset.\nUsing\nLog(w)\nfor the principal value of\nlog(w)\nand m, n as any integers the possible values of both sides are:\n• The identities\n(bc)x = bxcx\nand\n(b/c)x = bx/cx\nare valid when b and c are positive real numbers and x is a real number.\nBut, for the principal values, one has\nand\nOn the other hand, when x is an integer, the identities are valid for all nonzero complex numbers.\nIf exponentiation is considered as a multivalued function then the possible values of\n(−1 ⋅ −1)1/2\nare\n. The identity holds, but saying\n{1} =\nis incorrect.\n• The identity\n(ex)y = exy\nholds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:\nFor any integer n, we have:\nbut this is false when the integer n is nonzero.\nThe error is the following: by definition, e^y is a notation for \\exp(y), a true function, and x^y is a notation for which is a multi-valued function.\nThus the notation is ambiguous when\nx = e\n. Here, before expanding the exponent, the second line should be\nTherefore, when expanding the exponent, one has implicitly supposed that for complex values of z, which is wrong, as the complex logarithm is multivalued.\nIn other words, the wrong identity\n(ex)y = exy\nmust be replaced by the identity\nwhich is a true identity between multivalued functions.\n\n# Irrationality and transcendence\nIf b is a positive real [algebraic number](https://bliptext.com/articles/algebraic-number), and x is a rational number, then\nbx\nis an algebraic number.\nThis results from the theory of [algebraic extensions](https://bliptext.com/articles/algebraic-extension).\nThis remains true if b is any algebraic number, in which case, all values of\nbx\n(as a [multivalued function](https://bliptext.com/articles/multivalued-function)) are algebraic.\nIf x is [irrational](https://bliptext.com/articles/irrational-number) (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of\nbx\nare [transcendental](https://bliptext.com/articles/transcendental-number) (that is, not algebraic), except if b equals\n0\nor\n1\n.\nIn other words, if x is irrational and then at least one of b, x and\nbx\nis transcendental.\n\n# Integer powers in algebra\nThe definition of exponentiation with positive integer exponents as repeated multiplication may apply to any [associative operation](https://bliptext.com/articles/associative-operation) denoted as a multiplication.\nThe definition of\nx0\nrequires further the existence of a [multiplicative identity](https://bliptext.com/articles/multiplicative-identity).\nAn [algebraic structure](https://bliptext.com/articles/algebraic-structure) consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by\n1\nis a [monoid](https://bliptext.com/articles/monoid).\nIn such a monoid, exponentiation of an element x is defined inductively by\nIf n is a negative integer, x^n is defined only if x has a [multiplicative inverse](https://bliptext.com/articles/multiplicative-inverse).\nIn this case, the inverse of x is denoted\nx−1\n, and\nxn\nis defined as\nExponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:\nThese definitions are widely used in many areas of mathematics, notably for [groups](https://bliptext.com/articles/group-mathematics), [rings](https://bliptext.com/articles/ring-mathematics), [fields](https://bliptext.com/articles/field-mathematics), [square matrices](https://bliptext.com/articles/square-matrix) (which form a ring).\nThey apply also to [functions](https://bliptext.com/articles/function-mathematics) from a [set](https://bliptext.com/articles/set-mathematics) to itself, which form a monoid under [function composition](https://bliptext.com/articles/function-composition).\nThis includes, as specific instances, [geometric transformations](https://bliptext.com/articles/geometric-transformation), and [endomorphisms](https://bliptext.com/articles/endomorphism) of any [mathematical structure](https://bliptext.com/articles/mathematical-structure).\nWhen 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.\nFor example, if f is a [real function](https://bliptext.com/articles/real-function) whose valued can be multiplied, f^n denotes the exponentiation with respect of multiplication, and f^{\\circ n} may denote exponentiation with respect of [function composition](https://bliptext.com/articles/function-composition).\nThat is,\nand\nCommonly, (f^n)(x) is denoted f(x)^n, while is denoted f^n(x).\n\n## In a group\nA [multiplicative group](https://bliptext.com/articles/multiplicative-group) is a set with as [associative operation](https://bliptext.com/articles/associative-operation) denoted as multiplication, that has an [identity element](https://bliptext.com/articles/identity-element), and such that every element has an inverse.\nSo, if G is a group, x^n is defined for every x\\in G and every integer n.\nThe set of all powers of an element of a group form a [subgroup](https://bliptext.com/articles/subgroup).\nA group (or subgroup) that consists of all powers of a specific element x is the [cyclic group](https://bliptext.com/articles/cyclic-group) generated by x. If all the powers of x are distinct, the group is [isomorphic](https://bliptext.com/articles/isomorphic) to the [additive group](https://bliptext.com/articles/additive-group) \\Z of the integers.\nOtherwise, the cyclic group is [finite](https://bliptext.com/articles/finite-group) (it has a [finite](https://bliptext.com/articles/finite-group) number of elements), and its number of elements is the [order](https://bliptext.com/articles/order-group-theory) of x. If the [order](https://bliptext.com/articles/order-group-theory) of x is n, then x^n=x^0=1, and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent\n0\nor\n1\n).\nOrder of elements play a fundamental role in [group theory](https://bliptext.com/articles/group-theory).\nFor 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).\nThe possible orders of group elements are important in the study of the structure of a group (see [Sylow theorems](https://bliptext.com/articles/sylow-theorems)), and in the [classification of finite simple groups](https://bliptext.com/articles/classification-of-finite-simple-groups).\nSuperscript notation is also used for [conjugation](https://bliptext.com/articles/conjugacy-class); that is,\ngh = h−1gh\n, where\ng\nand\nh\nare elements of a group.\nThis notation cannot be confused with exponentiation, since the superscript is not an integer.\nThe motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely and\n\n## In a ring\nIn a [ring](https://bliptext.com/articles/ring-mathematics), it may occur that some nonzero elements satisfy x^n=0 for some integer n. Such an element is said to be [nilpotent](https://bliptext.com/articles/nilpotent).\nIn a [commutative ring](https://bliptext.com/articles/commutative-ring), the nilpotent elements form an [ideal](https://bliptext.com/articles/ideal-ring-theory), called the [nilradical](https://bliptext.com/articles/nilradical-of-a-ring) of the ring.\nIf the nilradical is [reduced](https://bliptext.com/articles/reduced-ring) to the [zero ideal](https://bliptext.com/articles/zero-ideal) (that is, if x\\neq 0 implies x^n\\neq 0 for every positive integer n), the commutative ring is said to be [reduced](https://bliptext.com/articles/reduced-ring).\nReduced rings are important in [algebraic geometry](https://bliptext.com/articles/algebraic-geometry), since the [coordinate ring](https://bliptext.com/articles/coordinate-ring) of an [affine algebraic set](https://bliptext.com/articles/affine-algebraic-set) is always a reduced ring.\nMore generally, given an ideal I in a commutative ring R, the set of the elements of R that have a power in I is an ideal, called the [radical](https://bliptext.com/articles/radical-of-an-ideal) of I. The nil[radical](https://bliptext.com/articles/radical-of-an-ideal) is the [radical](https://bliptext.com/articles/radical-of-an-ideal) of the [zero ideal](https://bliptext.com/articles/zero-ideal).\nA [radical ideal](https://bliptext.com/articles/radical-ideal) is an ideal that equals its own radical.\nIn a [polynomial ring](https://bliptext.com/articles/polynomial-ring) over a [field](https://bliptext.com/articles/field-mathematics) k, 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](https://bliptext.com/articles/hilbert-s-nullstellensatz)).\n\n## Matrices and linear operators\nIf\nA\nis a square matrix, then the product of\nA\nwith itself\nn\ntimes is called the [matrix power](https://bliptext.com/articles/matrix-power).\nAlso A^0 is defined to be the identity matrix, and if\nA\nis invertible, then.\nMatrix powers appear often in the context of [discrete dynamical systems](https://bliptext.com/articles/discrete-dynamical-system), where the matrix\nA\nexpresses a transition from a state vector\nx\nof some system to the next state\nAx\nof the system.\nThis is the standard interpretation of a [Markov chain](https://bliptext.com/articles/markov-chain), for example.\nThen A^2x is the state of the system after two time steps, and so forth: A^nx is the state of the system after\nn\ntime steps.\nThe matrix power A^n is the transition matrix between the state now and the state at a time\nn\nsteps in the future.\nSo computing matrix powers is equivalent to solving the evolution of the dynamical system.\nIn many cases, matrix powers can be expediently computed by using [eigenvalues and eigenvectors](https://bliptext.com/articles/eigenvalues-and-eigenvectors).\nApart from matrices, more general [linear operators](https://bliptext.com/articles/linear-operator) can also be exponentiated.\nAn example is the [derivative](https://bliptext.com/articles/derivative) operator of calculus, d/dx, which is a linear operator acting on functions f(x) to give a new function.\nThe\nn\nth power of the differentiation operator is the\nn\nth derivative:\nThese 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.\nThis is the starting point of the mathematical theory of [semigroups](https://bliptext.com/articles/c0-semigroup).\nJust as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics.\nExamples include approaches to solving the [heat equation](https://bliptext.com/articles/heat-equation), [Schrödinger equation](https://bliptext.com/articles/schr-dinger-equation), [wave equation](https://bliptext.com/articles/wave-equation), and other partial differential equations including a time evolution.\nThe special case of exponentiating the derivative operator to a non-integer power is called the [fractional derivative](https://bliptext.com/articles/fractional-derivative) which, together with the [fractional integral](https://bliptext.com/articles/fractional-integral), is one of the basic operations of the [fractional calculus](https://bliptext.com/articles/fractional-calculus).\n\n## Finite fields\nA [field](https://bliptext.com/articles/field-mathematics) is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is [associative](https://bliptext.com/articles/associative) and every nonzero element has a [multiplicative inverse](https://bliptext.com/articles/multiplicative-inverse).\nThis implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of\n0\n. Common examples are the field of [complex numbers](https://bliptext.com/articles/complex-number), the [real numbers](https://bliptext.com/articles/real-number) and the [rational numbers](https://bliptext.com/articles/rational-number), considered earlier in this article, which are all [infinite](https://bliptext.com/articles/infinite-set).\nA finite field is a field with a [finite number](https://bliptext.com/articles/finite-set) of elements.\nThis number of elements is either a [prime number](https://bliptext.com/articles/prime-number) or a [prime power](https://bliptext.com/articles/prime-power); that is, it has the form q=p^k, where p is a [prime number](https://bliptext.com/articles/prime-number), and k is a positive integer.\nFor every such q, there are fields with q elements.\nThe fields with q elements are all [isomorphic](https://bliptext.com/articles/isomorphic), which allows, in general, working as if there were only one field with q elements, denoted\nOne has\nfor every\nA [primitive element](https://bliptext.com/articles/primitive-element-finite-field) in \\mathbb F_q is an element g such that the set of the\nq − 1\nfirst powers of g (that is, ) equals the set of the nonzero elements of There are primitive elements in where \\varphi is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function).\nIn the [freshman's dream](https://bliptext.com/articles/freshman-s-dream) identity\nis true for the exponent p. As x^p=x in It follows that the map\nis [linear](https://bliptext.com/articles/linear-map) over and is a [field automorphism](https://bliptext.com/articles/field-automorphism), called the [Frobenius automorphism](https://bliptext.com/articles/frobenius-automorphism).\nIf q=p^k, the field \\mathbb F_q has k automorphisms, which are the k first powers (under [composition](https://bliptext.com/articles/function-composition)) of F. In other words, the [Galois group](https://bliptext.com/articles/galois-group) of \\mathbb F_q is [cyclic](https://bliptext.com/articles/cyclic-group) of order k, generated by the Frobenius automorphism.\nThe [Diffie–Hellman key exchange](https://bliptext.com/articles/diffie-hellman-key-exchange) is an application of exponentiation in finite fields that is widely used for [secure communications](https://bliptext.com/articles/secure-communication).\nIt uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the [discrete logarithm](https://bliptext.com/articles/discrete-logarithm), is computationally expensive.\nMore precisely, if g is a primitive element in then g^e can be efficiently computed with [exponentiation by squaring](https://bliptext.com/articles/exponentiation-by-squaring) for any e, even if q is large, while there is no known computationally practical algorithm that allows retrieving e from g^e if q is sufficiently large.\n\n# Powers of sets\nThe [Cartesian product](https://bliptext.com/articles/cartesian-product) of two [sets](https://bliptext.com/articles/set-mathematics) S and T is the set of the [ordered pairs](https://bliptext.com/articles/ordered-pair) (x,y) such that x\\in S and y\\in T. This operation is not properly [commutative](https://bliptext.com/articles/commutative) nor [associative](https://bliptext.com/articles/associative), but has these properties [up to](https://bliptext.com/articles/up-to) [canonical](https://bliptext.com/articles/canonical-map) [isomorphisms](https://bliptext.com/articles/isomorphism), that allow identifying, for example, (x,(y,z)), ((x,y),z), and (x,y,z).\nThis allows defining the nth power S^n of a set S as the set of all n-[tuples](https://bliptext.com/articles/tuple) of elements of S.\nWhen S is endowed with some structure, it is frequent that S^n is naturally endowed with a similar structure.\nIn this case, the term \"[direct product](https://bliptext.com/articles/direct-product)\" is generally used instead of \"Cartesian product\", and exponentiation denotes product structure.\nFor example \\R^n (where \\R denotes the real numbers) denotes the Cartesian product of n copies of \\R, as well as their direct product as [vector space](https://bliptext.com/articles/vector-space), [topological spaces](https://bliptext.com/articles/topological-space), [rings](https://bliptext.com/articles/ring-mathematics), etc.\n\n## Sets as exponents\nA n-tuple of elements of S can be considered as a [function](https://bliptext.com/articles/function-mathematics) from This generalizes to the following notation.\nGiven two sets S and T, the set of all functions from T to S is denoted S^T.\nThis exponential notation is justified by the following canonical isomorphisms (for the first one, see [Currying](https://bliptext.com/articles/currying)):\nwhere \\times denotes the Cartesian product, and \\sqcup the [disjoint union](https://bliptext.com/articles/disjoint-union).\nOne can use sets as exponents for other operations on sets, typically for [direct sums](https://bliptext.com/articles/direct-sum) of [abelian groups](https://bliptext.com/articles/abelian-group), [vector spaces](https://bliptext.com/articles/vector-space), or [modules](https://bliptext.com/articles/module-mathematics).\nFor distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses.\nFor example, \\R^\\N denotes the vector space of the [infinite sequences](https://bliptext.com/articles/infinite-sequence) of real numbers, and \\R^{(\\N)} the vector space of those sequences that have a finite number of nonzero elements.\nThe latter has a [basis](https://bliptext.com/articles/basis-linear-algebra) consisting of the sequences with exactly one nonzero element that equals\n1\n, while the [Hamel bases](https://bliptext.com/articles/hamel-basis) of the former cannot be explicitly described (because their existence involves [Zorn's lemma](https://bliptext.com/articles/zorn-s-lemma)).\nIn this context,\n2\ncan represents the set \\{0,1\\}.\nSo, 2^S denotes the [power set](https://bliptext.com/articles/power-set) of S, that is the set of the functions from S to \\{0,1\\}, which can be identified with the set of the [subsets](https://bliptext.com/articles/subset) of S, by mapping each function to the [inverse image](https://bliptext.com/articles/inverse-image) of\n1\n.\nThis fits in with the [exponentiation of cardinal numbers](https://bliptext.com/articles/cardinal-exponentiation), in the sense that\n=\n, where\nis the cardinality of\nX\n.\n\n## In category theory\nIn the [category of sets](https://bliptext.com/articles/category-of-sets), the [morphisms](https://bliptext.com/articles/morphism) between sets X and Y are the functions from X to Y. It results that the set of the functions from X to Y that is denoted Y^X in the preceding section can also be denoted \\hom(X,Y).\nThe isomorphism can be rewritten\nThis means the functor \"exponentiation to the power T\" is a [right adjoint](https://bliptext.com/articles/right-adjoint) to the functor \"direct product with T\".\nThis generalizes to the definition of [exponentiation in a category](https://bliptext.com/articles/exponential-category-theory) in which finite [direct products](https://bliptext.com/articles/direct-product) exist: in such a category, the functor X\\to X^T is, if it exists, a right adjoint to the functor A category is called a Cartesian closed category, if [direct products](https://bliptext.com/articles/direct-product) exist, and the functor has a right adjoint for every T.\n\n# Repeated exponentiation\nJust as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or [tetration](https://bliptext.com/articles/tetration).\nIterating tetration leads to another operation, and so on, a concept named [hyperoperation](https://bliptext.com/articles/hyperoperation).\nThis sequence of operations is expressed by the [Ackermann function](https://bliptext.com/articles/ackermann-function) and [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation).\nJust as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation.\nEvaluated at\n(3, 3)\n, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7,625,597,484,987 (\n=327 = 33 3 = 33\n) respectively.\n\n# Limits of powers\n[Zero to the power of zero](https://bliptext.com/articles/zero-to-the-power-of-zero) gives a number of examples of limits that are of the [indeterminate form](https://bliptext.com/articles/indeterminate-form) 00.\nThe limits in these examples exist, but have different values, showing that the two-variable function\nxy\nhas no limit at the point\n(0, 0)\n. One may consider at what points this function does have a limit.\nMore precisely, consider the function defined on.\nThen\nD\ncan be viewed as a subset of\nR2\n(that is, the set of all pairs\n(x, y)\nwith\nx\n,\ny\nbelonging to the [extended real number line](https://bliptext.com/articles/extended-real-number-line)\nR = [−∞, +∞]\n, endowed with the [product topology](https://bliptext.com/articles/product-topology)), which will contain the points at which the function\nf\nhas a limit.\nIn fact,\nf\nhas a limit at all [accumulation points](https://bliptext.com/articles/accumulation-point) of\nD\n, except for\n(0, 0)\n,\n(+∞, 0)\n,\n(1, +∞)\nand\n(1, −∞)\n. Accordingly, this allows one to define the powers\nxy\nby continuity whenever\n0 ≤ x ≤ +∞\n,\n−∞ ≤ y ≤ +∞\n, except for\n00\n,\n(+∞)0\n,\n1+∞\nand\n1−∞\n, which remain indeterminate forms.\nUnder this definition by continuity, we obtain:\nx+∞ = +∞\nand\nx−∞ = 0\n, when\n1 < x ≤ +∞\n.\nx+∞ = 0\nand\nx−∞ = +∞\n, when\n0 ≤ x < 1\n.\n0y = 0\nand\n(+∞)y = +∞\n, when\n0 < y ≤ +∞\n.\n0y = +∞\nand\n(+∞)y = 0\n, when\n−∞ ≤ y < 0\n.\nThese powers are obtained by taking limits of\nxy\nfor positive values of\nx\n. This method does not permit a definition of\nxy\nwhen\nx < 0\n, since pairs\n(x, y)\nwith\nx < 0\nare not accumulation points of\nD\n.\nOn the other hand, when\nn\nis an integer, the power\nxn\nis already meaningful for all values of\nx\n, including negative ones.\nThis may make the definition\n0n = +∞\nobtained above for negative\nn\nproblematic when\nn\nis odd, since in this case\nxn → +∞\nas\nx\ntends to\n0\nthrough positive values, but not negative ones.\n\n# Efficient computation with integer exponents\nComputing\nbn\nusing iterated multiplication requires\nn − 1\nmultiplication operations, but it can be computed more efficiently than that, as illustrated by the following example.\nTo compute\n2100\n, apply [Horner's rule](https://bliptext.com/articles/horner-s-rule) to the exponent 100 written in binary:\nThen compute the following terms in order, reading Horner's rule from right to left.\nThis series of steps only requires 8 multiplications instead of 99.\nIn general, the number of multiplication operations required to compute\nbn\ncan be reduced to by using [exponentiation by squaring](https://bliptext.com/articles/exponentiation-by-squaring), where \\sharp n denotes the number of\n1\ns in the [binary representation](https://bliptext.com/articles/binary-representation) of n. 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](https://bliptext.com/articles/addition-chain-exponentiation).\nFinding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for\nbn\nis a difficult problem, for which no efficient algorithms are currently known (see [Subset sum problem](https://bliptext.com/articles/subset-sum-problem)), but many reasonably efficient heuristic algorithms are available.\nHowever, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.\n\n# Iterated functions\n[Function composition](https://bliptext.com/articles/function-composition) is a [binary operation](https://bliptext.com/articles/binary-operation) that is defined on [functions](https://bliptext.com/articles/function-mathematics) such that the [codomain](https://bliptext.com/articles/co[domain](https://bliptext.com/articles/domain-of-a-function)) of the function written on the right is included in the [domain](https://bliptext.com/articles/domain-of-a-function) of the function written on the left.\nIt is denoted g\\circ f, and defined as for every x in the domain of f.\nIf the domain of a function f equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the nth power of the function under composition, commonly called the nth iterate of the function.\nThus f^n denotes generally the nth iterate of f; for example, f^3(x) means f(f(f(x))).\nWhen a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the [pointwise multiplication](https://bliptext.com/articles/pointwise-multiplication), which induces another exponentiation.\nWhen using [functional notation](https://bliptext.com/articles/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.\nThus and When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example and For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the [trigonometric functions](https://bliptext.com/articles/trigonometric-functions).\nSo, \\sin^2 x and \\sin^2(x) both mean and not which, in any case, is rarely considered.\nHistorically, several variants of these notations were used by different authors.\nIn this context, the exponent -1 denotes always the [inverse function](https://bliptext.com/articles/inverse-function), if it exists.\nSo For the [multiplicative inverse](https://bliptext.com/articles/multiplicative-inverse) fractions are generally used as in\n\n# In programming languages\n[Programming languages](https://bliptext.com/articles/programming-language) generally express exponentiation either as an infix [operator](https://bliptext.com/articles/operator-computer-programming) or as a function application, as they do not support superscripts.\nThe most common operator symbol for exponentiation is the [caret](https://bliptext.com/articles/caret).\nThe [original version of ASCII](https://bliptext.com/articles/ascii) included an uparrow symbol, intended for exponentiation, but this was [replaced by the caret](https://bliptext.com/articles/caret) in 1967, so the caret became usual in programming languages.\nThe notations include:\nIn most programming languages with an infix exponentiation operator, it is [right-associative](https://bliptext.com/articles/operator-associativity), that is, is interpreted as.\nThis is because is equal to and thus not as useful.\nIn some languages, it is left-associative, notably in [Algol](https://bliptext.com/articles/algol), [MATLAB](https://bliptext.com/articles/matlab), and the [Microsoft Excel](https://bliptext.com/articles/microsoft-office-excel) formula language.\nOther programming languages use functional notation:\nStill others only provide exponentiation as part of standard [libraries](https://bliptext.com/articles/library-computing):\nIn some [statically typed](https://bliptext.com/articles/type-system) languages that prioritize [type safety](https://bliptext.com/articles/type-safety) such as [Rust](https://bliptext.com/articles/rust-programming-language), exponentiation is performed via a multitude of methods:","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:55:08.090Z","updated_at":"2025-01-25T17:55:08.090Z","search_vector":"'exponenti':1","search_content":" \n**Mark:** bn \n**Name:** notation \n**Variant 1 Caption:** base b and exponent n \n \n\n# \n[[File:Expo02.svg|thumb|315px|Graphs of\ny = bx\nfor various bases\nb\nEach curve passes through the point\n(0, 1)\nbecause any nonzero number raised to the power of\n0\nis\n1\n. At\nx = 1\n, the value of\ny\nequals the base because any number raised to the power of\n1\nis the number itself.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), **exponentiation** is an [operation](https://bliptext.com/articles/operation-mathematics) involving [two numbers](https://bliptext.com/articles/binary-operation): the [base](https://bliptext.com/articles/base-exponentiation) and the exponent or power.\nExponentiation is written as\nbn\n, where b is the base and n is the power; often said as \"b to the power n\".\nWhen n is a positive [integer](https://bliptext.com/articles/integer), exponentiation corresponds to repeated [multiplication](https://bliptext.com/articles/multiplication) of the base: that is,\nbn\nis the [product](https://bliptext.com/articles/product-mathematics) of multiplying n bases:\nIn particular, b^1=b.\nThe exponent is usually shown as a [superscript](https://bliptext.com/articles/superscript) to the right of the base as\nbn\nor in computer code as b^n, and may also be called \"b raised to the nth power\", \"b to the power of n\", \"the nth power of b\", or most briefly \"b to the n\".\nThe above definition of b^n immediately implies several properties, in particular the multiplication rule:\nThat is, when multiplying a base raised to one power times the same base raised to another power, the powers add.\nExtending this rule to the power zero gives, and dividing both sides by b^n gives.\nThat is, the multiplication rule implies the definition b^0=1.A similar argument implies the definition for negative integer powers: That is, extending the multiplication rule gives.\nDividing both sides by b^n gives.\nThis also implies the definition for fractional powers: For example,, meaning , which is the definition of square root:.\nThe definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define b^x for any positive real base b and any real number exponent x. More involved definitions allow [complex](https://bliptext.com/articles/complex-numbers) base and exponent, as well as certain types of [matrices](https://bliptext.com/articles/matrix-mathematics) as base or exponent.\nExponentiation is used extensively in many fields, including [economics](https://bliptext.com/articles/economics), [biology](https://bliptext.com/articles/biology), [chemistry](https://bliptext.com/articles/chemistry), [physics](https://bliptext.com/articles/physics), and [computer science](https://bliptext.com/articles/computer-science), with applications such as [compound interest](https://bliptext.com/articles/compound-interest), [population growth](https://bliptext.com/articles/population-growth), [chemical reaction kinetics](https://bliptext.com/articles/chemical-reaction-kinetics), [wave](https://bliptext.com/articles/wave) behavior, and [public-key cryptography](https://bliptext.com/articles/public-key-cryptography).\n\n# Etymology\nThe term exponent originates from the [Latin](https://bliptext.com/articles/latin) exponentem, the [present participle](https://bliptext.com/articles/present-participle) of exponere, meaning \"to put forth\".\nThe term power is a mistranslation of the [ancient [Greek](https://bliptext.com/articles/greek-mathematics)](https://bliptext.com/articles/ancient-greek) δύναμις (dúnamis, here: \"amplification\" ) used by the [Greek](https://bliptext.com/articles/greek-mathematics) mathematician [Euclid](https://bliptext.com/articles/euclid) for the square of a line, following [Hippocrates of Chios](https://bliptext.com/articles/hippocrates-of-chios).\n\n# History\n\n## Antiquity\n\n### The Sand Reckoner\nIn [The Sand Reckoner](https://bliptext.com/articles/the-sand-reckoner), [Archimedes](https://bliptext.com/articles/archimedes) proved the law of exponents,\n10a · 10b = 10a+b\n, necessary to manipulate powers of\n10\n. He then used powers of\n10\nto estimate the number of grains of sand that can be contained in the universe.\n\n## Islamic Golden Age\n\n### Māl and kaʿbah (\"square\" and \"cube\")\nIn the 9th century, the Persian mathematician [Al-Khwarizmi](https://bliptext.com/articles/al-khwarizmi) used the terms مَال (māl, \"possessions\", \"property\") for a [square](https://bliptext.com/articles/square-algebra)—the Muslims, \"like most mathematicians of those and earlier times, thought of a [square](https://bliptext.com/articles/square-algebra)d number as a depiction of an area, especially of land, hence property\" —and كَعْبَة ([Kaʿbah](https://bliptext.com/articles/kaaba), \"[cube](https://bliptext.com/articles/cube-algebra)\") for a [cube](https://bliptext.com/articles/cube-algebra), which later [Islamic](https://bliptext.com/articles/mathematics-in-the-medieval-islamic-world) mathematicians represented in [mathematical notation](https://bliptext.com/articles/mathematical-notation) as the letters [mīm](https://bliptext.com/articles/m-m) (m) and [kāf](https://bliptext.com/articles/k-f) (k), respectively, by the 15th century, as seen in the work of [Abu'l-Hasan ibn Ali al-Qalasadi](https://bliptext.com/articles/abu-l-hasan-ibn-ali-al-qalasadi).\n\n## 15th–18th century\n\n### Introducing exponents\n[Nicolas Chuquet](https://bliptext.com/articles/nicolas-chuquet) used a form of exponential notation in the 15th century, for example\n122\nto represent\n12x2\n. This was later used by [Henricus Grammateus](https://bliptext.com/articles/henricus-grammateus) and [Michael Stifel](https://bliptext.com/articles/michael-stifel) in the 16th century.\nIn the late 16th century, [Jost Bürgi](https://bliptext.com/articles/jost-b-rgi) would use Roman numerals for exponents in a way similar to that of Chuquet, for example iii 4 for\n4x3\n.\n\n### \"Exponent\"; \"square\" and \"cube\"\nThe word exponent was coined in 1544 by Michael Stifel.\nIn the 16th century, [Robert Recorde](https://bliptext.com/articles/robert-recorde) used the terms square, cube, zenzizenzic ([fourth power](https://bliptext.com/articles/fourth-power)), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and [zenzizenzizenzic](https://bliptext.com/articles/zenzizenzizenzic) (eighth).\nBiquadrate has been used to refer to the fourth power as well.\n\n### Modern exponential notation\nIn 1636, [James Hume](https://bliptext.com/articles/james-hume-mathematician) used in essence modern notation, when in L'algèbre de Viète he wrote\nAiii\nfor\nA3\n. Early in the 17th century, the first form of our modern exponential notation was introduced by [René Descartes](https://bliptext.com/articles/ren-descartes) in his text titled [La Géométrie](https://bliptext.com/articles/la-g-om-trie); there, the notation is introduced in Book I.\n\"I designate ...\naa\n, or\na2\nin multiplying\na\nby itself; and\na3\nin multiplying it once more again by\na\n, and thus to infinity.\"\nSome mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication.\nThus they would write [polynomials](https://bliptext.com/articles/polynomial), for example, as\nax + bxx + cx3 + d\n.\n\n### \"Indices\"\n[Samuel Jeake](https://bliptext.com/articles/samuel-jeake) introduced the term indices in 1696.\nThe term involution was used synonymously with the term indices, but had declined in usage and should not be confused with [its more common meaning](https://bliptext.com/articles/involution-mathematics).\n\n### Variable exponents, non-integer exponents\nIn 1748, [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) introduced variable exponents, and, implicitly, non-integer exponents by writing:\"Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant.\"\n\n## 20th century\nAs calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation.\nFor example [Konrad Zuse](https://bliptext.com/articles/konrad-zuse) introduced [floating point arithmetic](https://bliptext.com/articles/floating-point-arithmetic) in his 1938 computer Z1.\nOne [register](https://bliptext.com/articles/register-computer) contained representation of leading digits, and a second contained representation of the exponent of 10.\nEarlier [Leonardo Torres Quevedo](https://bliptext.com/articles/leonardo-torres-quevedo) contributed Essays on Automation (1914) which had suggested the floating-point representation of numbers.\nThe more flexible [decimal floating-point](https://bliptext.com/articles/decimal-floating-point) representation was introduced in 1946 with a [Bell Laboratories](https://bliptext.com/articles/bell-laboratories) computer.\nEventually educators and engineers adopted [scientific notation](https://bliptext.com/articles/scientific-notation) of numbers, consistent with common reference to [order of magnitude](https://bliptext.com/articles/order-of-magnitude) in a [ratio scale](https://bliptext.com/articles/ratio-scale).\n\n# Terminology\nThe expression\nb2 = b · b\nis called \"the [square](https://bliptext.com/articles/square-algebra) of\nb\n\" or \"\nb\nsquared\", because the area of a square with side-length\nb\nis\nb2\n. (It is true that it could also be called \"\nb\nto the second power\", but \"the square of\nb\n\" and \"\nb\nsquared\" are more traditional)\nSimilarly, the expression\nb3 = b · b · b\nis called \"the [cube](https://bliptext.com/articles/cube-algebra) of\nb\n\" or \"\nb\ncubed\", because the volume of a cube with side-length\nb\nis\nb3\n.\nWhen an exponent is a [positive integer](https://bliptext.com/articles/positive-integer), that exponent indicates how many copies of the base are multiplied together.\nFor example,\n35 = 3 · 3 · 3 · 3 · 3 = 243\n. The base\n3\nappears\n5\ntimes in the multiplication, because the exponent is\n5\n. Here,\n243\nis the 5th power of 3, or 3 raised to the 5th power.\nThe word \"raised\" is usually omitted, and sometimes \"power\" as well, so\n35\ncan be simply read \"3 to the 5th\", or \"3 to the 5\".\n\n# Integer exponents\nThe exponentiation operation with integer exponents may be defined directly from elementary [arithmetic operations](https://bliptext.com/articles/arithmetic-operation).\n\n## Positive exponents\nThe definition of the exponentiation as an iterated multiplication can be [formalized](https://bliptext.com/articles/formal-proof) by using [induction](https://bliptext.com/articles/mathematical-induction), and this definition can be used as soon as one has an [associative](https://bliptext.com/articles/associativity) multiplication:\nThe base case is and the [recurrence](https://bliptext.com/articles/recurrence-relation) is\nThe associativity of multiplication implies that for any positive integers m and n, and\n\n## Zero exponent\nAs mentioned earlier, a (nonzero) number raised to the\n0\npower is\n1\nThis value is also obtained by the [empty product](https://bliptext.com/articles/empty-product) convention, which may be used in every [algebraic structure](https://bliptext.com/articles/algebraic-structure) with a multiplication that has an [identity](https://bliptext.com/articles/multiplicative-identity).\nThis way the formula\nalso holds for n=0.\nThe case of\n00\nis controversial.\nIn contexts where only integer powers are considered, the value\n1\nis generally assigned to\n00\nbut, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.\n\n## Negative exponents\nExponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b: Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (\\infty).\nThis definition of exponentiation with negative exponents is the only one that allows extending the identity to negative exponents (consider the case m=-n).\nThe same definition applies to [invertible elements](https://bliptext.com/articles/invertible-element) in a multiplicative [monoid](https://bliptext.com/articles/monoid), that is, an [algebraic structure](https://bliptext.com/articles/algebraic-structure), with an associative multiplication and a [multiplicative identity](https://bliptext.com/articles/multiplicative-identity) denoted\n1\n(for example, the [square matrices](https://bliptext.com/articles/square-matrix) of a given dimension).\nIn particular, in such a structure, the inverse of an [invertible element](https://bliptext.com/articles/invertible-element) x is standardly denoted x^{-1}.\n\n## Identities and properties\nThe following [identities](https://bliptext.com/articles/identity-mathematics), often called **exponent rules**, hold for all integer exponents, provided that the base is non-zero:\nUnlike addition and multiplication, exponentiation is not [commutative](https://bliptext.com/articles/commutative): for example, 2^3 = 8, but reversing the operands gives the different value 3^2=9.\nAlso unlike addition and multiplication, exponentiation is not [associative](https://bliptext.com/articles/associative): for example,\n(23)2 = 82 = 64\n, whereas\n2(3 2) = 29 = 512\n. Without parentheses, the conventional [order of operations](https://bliptext.com/articles/order-of-operations) for [serial exponentiation](https://bliptext.com/articles/serial-exponentiation) in superscript notation is top-down (or right-associative), not bottom-up (or left-associative).\nThat is,\nwhich, in general, is different from\n\n## Powers of a sum\nThe powers of a sum can normally be computed from the powers of the summands by the [binomial formula](https://bliptext.com/articles/binomial-formula)\nHowever, this formula is true only if the summands commute (i.e. that\nab = ba\n), which is implied if they belong to a [structure](https://bliptext.com/articles/algebraic-structure) that is [commutative](https://bliptext.com/articles/commutative-property). Otherwise, if a and b are, say, [square matrices](https://bliptext.com/articles/square-matrix) of the same size, this formula cannot be used. It follows that in [computer algebra](https://bliptext.com/articles/computer-algebra), many [algorithms](https://bliptext.com/articles/algorithm) involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose [computer algebra](https://bliptext.com/articles/computer-algebra) systems use a different notation (sometimes\n^^\ninstead of\n^\n) for exponentiation with non-commuting bases, which is then called **non-commutative exponentiation**.\n\n## Combinatorial interpretation\nFor nonnegative integers n and m, the value of\nnm\nis the number of [functions](https://bliptext.com/articles/function-mathematics) from a [set](https://bliptext.com/articles/set-mathematics) of m elements to a [set](https://bliptext.com/articles/set-mathematics) of n elements (see [cardinal exponentiation](https://bliptext.com/articles/cardinal-exponentiation)).\nSuch functions can be represented as m-[tuples](https://bliptext.com/articles/tuple) from an n-element set (or as m-letter words from an n-letter alphabet).\nSome examples for particular values of m and n are given in the following table:\n\n## Particular bases\n\n### Powers of ten\nIn the base ten ([decimal](https://bliptext.com/articles/decimal)) number system, integer powers of\n10\nare written as the digit\n1\nfollowed or preceded by a number of zeroes determined by the sign and magnitude of the exponent.\nFor example,\n= 1,000\nand\n= 0\n.\nExponentiation with base\n10\nis used in [scientific notation](https://bliptext.com/articles/scientific-notation) to denote large or small numbers.\nFor instance, 299,792,458 m/s (the [speed of light](https://bliptext.com/articles/speed-of-light) in vacuum, in [metres per second](https://bliptext.com/articles/metres-per-second)) can be written as 2.998 m/s and then [approximated](https://bliptext.com/articles/approximation) as 2.998 m/s.\n[SI prefixes](https://bliptext.com/articles/si-prefix) based on powers of\n10\nare also used to describe small or large quantities.\nFor example, the prefix [kilo](https://bliptext.com/articles/kilo) means\n= 1,000\n, so a kilometre is 1,000 metres.\n\n### Powers of two\nThe first negative powers of\n2\nhave special names: 2^{-1}is a [half](https://bliptext.com/articles/one-half); 2^{-2} is a [quarter](https://bliptext.com/articles/4-number).\nPowers of\n2\nappear in [set theory](https://bliptext.com/articles/set-theory), since a set with\nn\nmembers has a [power set](https://bliptext.com/articles/power-set), the set of all of its [subsets](https://bliptext.com/articles/subset), which has\n2n\nmembers.\nInteger powers of\n2\nare important in [computer science](https://bliptext.com/articles/computer-science).\nThe positive integer powers\n2n\ngive the number of possible values for an\nn\n-[bit](https://bliptext.com/articles/bit) integer [binary number](https://bliptext.com/articles/binary-number); for example, a [byte](https://bliptext.com/articles/byte) may take\n28 = 256\ndifferent values.\nThe [binary number system](https://bliptext.com/articles/binary-number-system) expresses any number as a sum of powers of\n2\n, and denotes it as a sequence of\n0\nand\n1\n, separated by a [binary point](https://bliptext.com/articles/binary-point), where\n1\nindicates a power of\n2\nthat appears in the sum; the exponent is determined by the place of this\n1\n1\non the left of the point (starting from\n0\n), and the negative exponents are determined by the rank on the right of the point.\n\n### Powers of one\nEvery power of one equals:\n1n = 1\n.\n\n### Powers of zero\nFor a positive exponent\nn > 0\n, the nth power of zero is zero:\n0n = 0\n. For a negative\\ exponent, is undefined.\nThe expression [[zero to the power of zero|\n00\n]] is either [defined as](https://bliptext.com/articles/indeterminate-form), or it is left undefined.\n\n### Powers of negative one\nSince a negative number times another negative is positive, we have: Because of this, powers of\n−1\nare useful for expressing alternating [sequences](https://bliptext.com/articles/sequence).\nFor a similar discussion of powers of the complex number\ni\n, see .\n\n## Large exponents\nThe [limit of a sequence](https://bliptext.com/articles/limit-of-a-sequence) of powers of a number greater than one diverges; in other words, the sequence grows without bound:\nbn → ∞\nas\nn → ∞\nwhen\nb > 1\nThis can be read as \"b to the power of n tends to [+∞](https://bliptext.com/articles/extended-real-number-line) as n tends to infinity when b is greater than one\".\nPowers of a number with [absolute value](https://bliptext.com/articles/absolute-value) less than one tend to zero:\nbn → 0\nas\nn → ∞\nwhen\nAny power of one is always one:\nbn = 1\nfor all\nn\nfor\nb = 1\nPowers of a negative number b\\leq -1 alternate between positive and negative as\nn\nalternates between even and odd, and thus do not tend to any limit as\nn\ngrows.\nIf the exponentiated number varies while tending to\n1\nas the exponent tends to infinity, then the limit is not necessarily one of those above.\nA particularly important case is\n(1 + 1/n)n → e\nas\nn → ∞\nSee below.\nOther limits, in particular those of expressions that take on an [indeterminate form](https://bliptext.com/articles/indeterminate-form), are described in below.\n\n## Power functions\n[[File:Potenssi 1 3 5.svg|thumb|left|Power functions for\nn = 1, 3, 5\n]] [[File:Potenssi 2 4 6.svg|thumb|Power functions for\nn = 2, 4, 6\n]]\nReal functions of the form f(x) = cx^n, where c \\ne 0, are sometimes called power functions.\nWhen n is an [integer](https://bliptext.com/articles/integer) and n \\ge 1, two primary families exist: for n even, and for n odd.\nIn general for c > 0, when n is even f(x) = cx^n will tend towards positive [infinity](https://bliptext.com/articles/infinity-mathematics) with increasing x, and also towards positive [infinity](https://bliptext.com/articles/infinity-mathematics) with decreasing x. All graphs from the family of even power functions have the general shape of y=cx^2, flattening more in the middle as n increases.\nFunctions with this kind of [symmetry](https://bliptext.com/articles/symmetry) are called [even functions](https://bliptext.com/articles/even-functions).\nWhen n is odd, f(x)'s [asymptotic](https://bliptext.com/articles/asymptotic) behavior reverses from positive x to negative x. For c > 0, f(x) = cx^n will also tend towards positive [infinity](https://bliptext.com/articles/infinity-mathematics) with increasing x, but towards negative [infinity](https://bliptext.com/articles/infinity-mathematics) with decreasing x. All graphs from the family of odd power functions have the general shape of y=cx^3, flattening more in the middle as n increases and losing all flatness there in the straight line for n=1.\nFunctions with this kind of symmetry are called [odd functions](https://bliptext.com/articles/odd-function).\nFor c 0 integers, then x^{p/q} is defined as The equality on the right may be derived by setting and writing\nIf r is a positive rational number,\n0r = 0\n, by definition.\nAll these definitions are required for extending the identity to rational exponents.\nOn the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers.\nFor example, a negative real number has a real nth root, which is negative, if n is [odd](https://bliptext.com/articles/odd-number), and no real root if n is even.\nIn the latter case, whichever complex nth root one chooses for x^\\frac 1n, the identity cannot be satisfied.\nFor example,\nSee **' and '** for details on the way these problems may be handled.\n\n# Real exponents\nFor positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (**', below), or in terms of the [logarithm](https://bliptext.com/articles/logarithm) of the base and the [exponential function](https://bliptext.com/articles/exponential-function) ('**, below).\nThe result is always a positive real number, and the [identities and properties](https://bliptext.com/articles/) shown above for integer exponents remain true with these definitions for real exponents.\nThe second definition is more commonly used, since it generalizes straightforwardly to [complex](https://bliptext.com/articles/complex-number) exponents.\nOn the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values.\nOne may choose one of these values, called the [principal value](https://bliptext.com/articles/principal-value), but there is no choice of the [principal value](https://bliptext.com/articles/principal-value) for which the identity\nis true; see .\nTherefore, exponentiation with a basis that is not a positive real number is generally viewed as a [multivalued function](https://bliptext.com/articles/multivalued-function).\n\n## Limits of rational exponents\n[[File:Continuity of the Exponential at 0.svg|thumb|The limit of\ne1/n\nis\ne0 = 1\nwhen n tends to the infinity.]]\nSince any [irrational number](https://bliptext.com/articles/irrational-number) can be expressed as the [limit of a sequence](https://bliptext.com/articles/limit-of-a-sequence) of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by [continuity](https://bliptext.com/articles/continuous-function) with the rule\nwhere the limit is taken over rational values of r only.\nThis limit exists for every positive b and every real x.\nFor example, if\nx = π\n, the [non-terminating decimal](https://bliptext.com/articles/non-terminating-decimal) representation\nπ = 3.14159...\nand the [monotonicity](https://bliptext.com/articles/monotone-function) of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain b^\\pi:\nSo, the upper bounds and the lower bounds of the intervals form two [sequences](https://bliptext.com/articles/sequence-mathematics) that have the same limit, denoted b^\\pi.\nThis defines b^x for every positive b and real x as a [continuous function](https://bliptext.com/articles/continuous-function) of b and x. See also [Well-defined expression](https://bliptext.com/articles/well-defined-expression).\n\n## Exponential function\nThe exponential function may be defined as where is [Euler's number](https://bliptext.com/articles/euler-s-number), but to avoid [circular reasoning](https://bliptext.com/articles/circular-reasoning), this definition cannot be used here.\nRather, we give an independent definition of the exponential function \\exp(x), and of e=\\exp(1), relying only on positive integer powers (repeated multiplication).\nThen we sketch the proof that this agrees with the previous defintion:\nThere are [many equivalent ways to define the exponential function](https://bliptext.com/articles/characterizations-of-the-exponential-function), one of them being\nOne has \\exp(0)=1, and the exponential identity (or multiplication rule) holds as well, since\nand the second-order term does not affect the limit, yielding.\nEuler's number can be defined as e=\\exp(1).\nIt follows from the preceding equations that \\exp(x)=e^x when x is an integer (this results from the repeated-multiplication definition of the exponentiation).\nIf x is real, \\exp(x)=e^x results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity of the exponential function otherwise.\nThe limit that defines the exponential function converges for every [complex](https://bliptext.com/articles/complex-number) value of x, and therefore it can be used to extend the definition of \\exp(z), and thus e^z, from the real numbers to any [complex](https://bliptext.com/articles/complex-number) argument z. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for [complex](https://bliptext.com/articles/complex-number) base and exponent.\n\n## Powers via logarithms\nThe definition of\nex\nas the exponential function allows defining\nbx\nfor every positive real numbers b, in terms of exponential and [logarithm](https://bliptext.com/articles/logarithm) function.\nSpecifically, the fact that the [natural logarithm](https://bliptext.com/articles/natural-logarithm)\nln(x)\nis the [inverse](https://bliptext.com/articles/inverse-function) of the exponential function\nex\nmeans that one has\nfor every\nb > 0\n. For preserving the identity one must have\nSo, e^{x \\ln b} can be used as an alternative definition of\nbx\nfor any positive real b. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.\n\n# Complex exponents with a positive real base\nIf b is a positive real number, exponentiation with base b and [complex](https://bliptext.com/articles/complex-number) exponent z is defined by means of the exponential function with [complex](https://bliptext.com/articles/complex-number) argument (see the end of , above) as where \\ln b denotes the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of b.\nThis satisfies the identity\nIn general,\nis not defined, since\nbz\nis not a real number.\nIf a meaning is given to the exponentiation of a complex number (see , below), one has, in general,\nunless z is real or t is an integer.\n[Euler's formula](https://bliptext.com/articles/euler-s-formula),\nallows expressing the [polar form](https://bliptext.com/articles/polar-form) of b^z in terms of the [real and imaginary parts](https://bliptext.com/articles/real-and-imaginary-parts) of z, namely\nwhere the [absolute value](https://bliptext.com/articles/absolute-value) of the [trigonometric](https://bliptext.com/articles/trigonometry) factor is one.\nThis results from\n\n# Non-integer powers of complex numbers\nIn the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only.\nFor other bases, difficulties appear already with the apparently simple case of nth roots, that is, of exponents 1/n, where n is a positive integer.\nAlthough the general theory of exponentiation with non-integer exponents applies to nth roots, this case deserves to be considered first, since it does not need to use [complex logarithms](https://bliptext.com/articles/complex-logarithm), and is therefore easier to understand.\n\n## nth roots of a complex number\nEvery nonzero complex number z may be written in [polar form](https://bliptext.com/articles/polar-form) as where \\rho is the [absolute value](https://bliptext.com/articles/absolute-value) of z, and \\theta is its [argument](https://bliptext.com/articles/argument-complex-analysis).\nThe argument is defined [up to](https://bliptext.com/articles/up-to) an integer multiple of\n2π\nThe polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments.\nIt follows that the polar form of an nth root of a complex number can be obtained by taking the nth root of the absolute value and dividing its argument by n:\nIf 2\\pi is added to \\theta, the complex number is not changed, but this adds 2i\\pi/n to the argument of the nth root, and provides a new nth root.\nThis can be done n times, and provides the n nth roots of the complex number.\nIt is usual to choose one of the n nth root as the [principal root](https://bliptext.com/articles/principal-root).\nThe common choice is to choose the nth root for which that is, the nth root that has the largest real part, and, if there are two, the one with positive imaginary part.\nThis makes the principal nth root a [continuous function](https://bliptext.com/articles/continuous-function) in the whole complex plane, except for negative real values of the [radicand](https://bliptext.com/articles/radicand).\nThis function equals the usual nth root for positive real radicands.\nFor negative real radicands, and odd exponents, the principal nth root is not real, although the usual nth root is real.\n[Analytic continuation](https://bliptext.com/articles/analytic-continuation) shows that the principal nth root is the unique [complex differentiable](https://bliptext.com/articles/complex-differentiable) function that extends the usual nth root to the complex plane without the nonpositive real numbers.\nIf the complex number is moved around zero by increasing its argument, after an increment of 2\\pi, the complex number comes back to its initial position, and its nth roots are [permuted circularly](https://bliptext.com/articles/circular-permutation) (they are multiplied by e^{2i\\pi/n}).\nThis shows that it is not possible to define a nth root function that is continuous in the whole complex plane.\n\n### Roots of unity\n[[File:One3Root.svg|thumb|right|The three third roots of\n1\n]] The nth roots of unity are the n complex numbers such that\nwn = 1\n, where n is a positive integer.\nThey arise in various areas of mathematics, such as in [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) or algebraic solutions of algebraic equations ([Lagrange resolvent](https://bliptext.com/articles/lagrange-resolvent)).\nThe n nth roots of unity are the n first powers of, that is The nth roots of unity that have this generating property are called primitive nth roots of unity; they have the form with k [coprime](https://bliptext.com/articles/coprime-integers) with n. The unique primitive square root of unity is -1; the primitive fourth roots of unity are i and -i.\nThe nth roots of unity allow expressing all nth roots of a complex number z as the n products of a given nth roots of z with a nth root of unity.\nGeometrically, the nth roots of unity lie on the [unit circle](https://bliptext.com/articles/unit-circle) of the [complex plane](https://bliptext.com/articles/complex-plane) at the vertices of a [regular n-gon](https://bliptext.com/articles/regular-polygon) with one vertex on the real number 1.\nAs the number is the primitive nth root of unity with the smallest positive [argument](https://bliptext.com/articles/argument-complex-analysis), it is called the principal primitive nth root of unity, sometimes shortened as principal nth root of unity, although this terminology can be confused with the [principal value](https://bliptext.com/articles/principal-value) of 1^{1/n}, which is 1.\n\n## Complex exponentiation\nDefining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for z^w.\nSo, either a [principal value](https://bliptext.com/articles/principal-value) is defined, which is not continuous for the values of z that are real and nonpositive, or z^w is defined as a [multivalued function](https://bliptext.com/articles/multivalued-function).\nIn all cases, the [complex logarithm](https://bliptext.com/articles/complex-logarithm) is used to define complex exponentiation as where \\log z is the variant of the [complex logarithm](https://bliptext.com/articles/complex-logarithm) that is used, which is a function or a [multivalued function](https://bliptext.com/articles/multivalued-function) such that\nfor every z in its [domain of definition](https://bliptext.com/articles/domain-of-a-function).\n\n### Principal value\nThe [principal value](https://bliptext.com/articles/principal-value) of the [complex logarithm](https://bliptext.com/articles/complex-logarithm) is the unique continuous function, commonly denoted \\log, such that, for every nonzero complex number z, and the [argument](https://bliptext.com/articles/argument-complex-analysis) of z satisfies\nThe principal value of the complex logarithm is not defined for z=0, it is [discontinuous](https://bliptext.com/articles/continuous-function) at negative real values of z, and it is [holomorphic](https://bliptext.com/articles/holomorphic) (that is, complex differentiable) elsewhere.\nIf z is real and positive, the principal value of the complex logarithm is the natural logarithm:\nThe principal value of z^w is defined as\nwhere \\log z is the principal value of the logarithm.\nThe function is holomorphic except in the neighbourhood of the points where z is real and nonpositive.\nIf z is real and positive, the principal value of z^w equals its usual value defined above.\nIf w=1/n, where n is an integer, this principal value is the same as the one defined above.\n\n### Multivalued function\nIn some contexts, there is a problem with the discontinuity of the principal values of \\log z and z^w at the negative real values of z. In this case, it is useful to consider these functions as [multivalued functions](https://bliptext.com/articles/multivalued-function).\nIf \\log z denotes one of the values of the multivalued logarithm (typically its principal value), the other values are where k is any integer.\nSimilarly, if z^w is one value of the exponentiation, then the other values are given by where k is any integer.\nDifferent values of k give different values of z^w unless w is a [rational number](https://bliptext.com/articles/rational-number), that is, there is an integer d such that dw is an integer.\nThis results from the [periodicity](https://bliptext.com/articles/periodic-function) of the exponential function, more specifically, that e^a=e^b if and only if a-b is an integer multiple of 2\\pi i.\nIf w=\\frac mn is a rational number with m and n [coprime integers](https://bliptext.com/articles/coprime-integers) with n>0, then z^w has exactly n values.\nIn the case m=1, these values are the same as those described in [§ nth roots of a complex number](https://bliptext.com/articles/).\nIf w is an integer, there is only one value that agrees with that of.\nThe multivalued exponentiation is holomorphic for z\\ne 0, in the sense that its [graph](https://bliptext.com/articles/graph-of-a-function) consists of several sheets that define each a holomorphic function in the neighborhood of every point.\nIf z varies continuously along a circle around\n0\n, then, after a turn, the value of z^w has changed of sheet.\n\n### Computation\nThe canonical form x+iy of z^w can be computed from the canonical form of z and w. Although this can be described by a single formula, it is clearer to split the computation in several steps.\n\n#### Examples\n−2\nis So, the above described method gives the values In this case, all the values have the same argument 4\\ln 2, and different absolute values.\nIn both examples, all values of z^w have the same argument.\nMore generally, this is true if and only if the [real part](https://bliptext.com/articles/real-part) of w is an integer.\n\n### Failure of power and logarithm identities\nSome identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions.\nFor example:\n• The identity\nlog(bx) = x ⋅ log b\nholds whenever b is a positive real number and x is a real number.\nBut for the principal branch of the complex logarithm one has\nRegardless of which branch of the logarithm is used, a similar failure of the identity will exist.\nThe best that can be said (if only using this result) is that:\nThis identity does not hold even when considering log as a multivalued function.\nThe possible values of\nlog(wz)\ncontain those of\nz ⋅ log w\nas a proper subset.\nUsing\nLog(w)\nfor the principal value of\nlog(w)\nand m, n as any integers the possible values of both sides are:\n• The identities\n(bc)x = bxcx\nand\n(b/c)x = bx/cx\nare valid when b and c are positive real numbers and x is a real number.\nBut, for the principal values, one has\nand\nOn the other hand, when x is an integer, the identities are valid for all nonzero complex numbers.\nIf exponentiation is considered as a multivalued function then the possible values of\n(−1 ⋅ −1)1/2\nare\n. The identity holds, but saying\n{1} =\nis incorrect.\n• The identity\n(ex)y = exy\nholds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:\nFor any integer n, we have:\nbut this is false when the integer n is nonzero.\nThe error is the following: by definition, e^y is a notation for \\exp(y), a true function, and x^y is a notation for which is a multi-valued function.\nThus the notation is ambiguous when\nx = e\n. Here, before expanding the exponent, the second line should be\nTherefore, when expanding the exponent, one has implicitly supposed that for complex values of z, which is wrong, as the complex logarithm is multivalued.\nIn other words, the wrong identity\n(ex)y = exy\nmust be replaced by the identity\nwhich is a true identity between multivalued functions.\n\n# Irrationality and transcendence\nIf b is a positive real [algebraic number](https://bliptext.com/articles/algebraic-number), and x is a rational number, then\nbx\nis an algebraic number.\nThis results from the theory of [algebraic extensions](https://bliptext.com/articles/algebraic-extension).\nThis remains true if b is any algebraic number, in which case, all values of\nbx\n(as a [multivalued function](https://bliptext.com/articles/multivalued-function)) are algebraic.\nIf x is [irrational](https://bliptext.com/articles/irrational-number) (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of\nbx\nare [transcendental](https://bliptext.com/articles/transcendental-number) (that is, not algebraic), except if b equals\n0\nor\n1\n.\nIn other words, if x is irrational and then at least one of b, x and\nbx\nis transcendental.\n\n# Integer powers in algebra\nThe definition of exponentiation with positive integer exponents as repeated multiplication may apply to any [associative operation](https://bliptext.com/articles/associative-operation) denoted as a multiplication.\nThe definition of\nx0\nrequires further the existence of a [multiplicative identity](https://bliptext.com/articles/multiplicative-identity).\nAn [algebraic structure](https://bliptext.com/articles/algebraic-structure) consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by\n1\nis a [monoid](https://bliptext.com/articles/monoid).\nIn such a monoid, exponentiation of an element x is defined inductively by\nIf n is a negative integer, x^n is defined only if x has a [multiplicative inverse](https://bliptext.com/articles/multiplicative-inverse).\nIn this case, the inverse of x is denoted\nx−1\n, and\nxn\nis defined as\nExponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:\nThese definitions are widely used in many areas of mathematics, notably for [groups](https://bliptext.com/articles/group-mathematics), [rings](https://bliptext.com/articles/ring-mathematics), [fields](https://bliptext.com/articles/field-mathematics), [square matrices](https://bliptext.com/articles/square-matrix) (which form a ring).\nThey apply also to [functions](https://bliptext.com/articles/function-mathematics) from a [set](https://bliptext.com/articles/set-mathematics) to itself, which form a monoid under [function composition](https://bliptext.com/articles/function-composition).\nThis includes, as specific instances, [geometric transformations](https://bliptext.com/articles/geometric-transformation), and [endomorphisms](https://bliptext.com/articles/endomorphism) of any [mathematical structure](https://bliptext.com/articles/mathematical-structure).\nWhen 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.\nFor example, if f is a [real function](https://bliptext.com/articles/real-function) whose valued can be multiplied, f^n denotes the exponentiation with respect of multiplication, and f^{\\circ n} may denote exponentiation with respect of [function composition](https://bliptext.com/articles/function-composition).\nThat is,\nand\nCommonly, (f^n)(x) is denoted f(x)^n, while is denoted f^n(x).\n\n## In a group\nA [multiplicative group](https://bliptext.com/articles/multiplicative-group) is a set with as [associative operation](https://bliptext.com/articles/associative-operation) denoted as multiplication, that has an [identity element](https://bliptext.com/articles/identity-element), and such that every element has an inverse.\nSo, if G is a group, x^n is defined for every x\\in G and every integer n.\nThe set of all powers of an element of a group form a [subgroup](https://bliptext.com/articles/subgroup).\nA group (or subgroup) that consists of all powers of a specific element x is the [cyclic group](https://bliptext.com/articles/cyclic-group) generated by x. If all the powers of x are distinct, the group is [isomorphic](https://bliptext.com/articles/isomorphic) to the [additive group](https://bliptext.com/articles/additive-group) \\Z of the integers.\nOtherwise, the cyclic group is [finite](https://bliptext.com/articles/finite-group) (it has a [finite](https://bliptext.com/articles/finite-group) number of elements), and its number of elements is the [order](https://bliptext.com/articles/order-group-theory) of x. If the [order](https://bliptext.com/articles/order-group-theory) of x is n, then x^n=x^0=1, and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent\n0\nor\n1\n).\nOrder of elements play a fundamental role in [group theory](https://bliptext.com/articles/group-theory).\nFor 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).\nThe possible orders of group elements are important in the study of the structure of a group (see [Sylow theorems](https://bliptext.com/articles/sylow-theorems)), and in the [classification of finite simple groups](https://bliptext.com/articles/classification-of-finite-simple-groups).\nSuperscript notation is also used for [conjugation](https://bliptext.com/articles/conjugacy-class); that is,\ngh = h−1gh\n, where\ng\nand\nh\nare elements of a group.\nThis notation cannot be confused with exponentiation, since the superscript is not an integer.\nThe motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely and\n\n## In a ring\nIn a [ring](https://bliptext.com/articles/ring-mathematics), it may occur that some nonzero elements satisfy x^n=0 for some integer n. Such an element is said to be [nilpotent](https://bliptext.com/articles/nilpotent).\nIn a [commutative ring](https://bliptext.com/articles/commutative-ring), the nilpotent elements form an [ideal](https://bliptext.com/articles/ideal-ring-theory), called the [nilradical](https://bliptext.com/articles/nilradical-of-a-ring) of the ring.\nIf the nilradical is [reduced](https://bliptext.com/articles/reduced-ring) to the [zero ideal](https://bliptext.com/articles/zero-ideal) (that is, if x\\neq 0 implies x^n\\neq 0 for every positive integer n), the commutative ring is said to be [reduced](https://bliptext.com/articles/reduced-ring).\nReduced rings are important in [algebraic geometry](https://bliptext.com/articles/algebraic-geometry), since the [coordinate ring](https://bliptext.com/articles/coordinate-ring) of an [affine algebraic set](https://bliptext.com/articles/affine-algebraic-set) is always a reduced ring.\nMore generally, given an ideal I in a commutative ring R, the set of the elements of R that have a power in I is an ideal, called the [radical](https://bliptext.com/articles/radical-of-an-ideal) of I. The nil[radical](https://bliptext.com/articles/radical-of-an-ideal) is the [radical](https://bliptext.com/articles/radical-of-an-ideal) of the [zero ideal](https://bliptext.com/articles/zero-ideal).\nA [radical ideal](https://bliptext.com/articles/radical-ideal) is an ideal that equals its own radical.\nIn a [polynomial ring](https://bliptext.com/articles/polynomial-ring) over a [field](https://bliptext.com/articles/field-mathematics) k, 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](https://bliptext.com/articles/hilbert-s-nullstellensatz)).\n\n## Matrices and linear operators\nIf\nA\nis a square matrix, then the product of\nA\nwith itself\nn\ntimes is called the [matrix power](https://bliptext.com/articles/matrix-power).\nAlso A^0 is defined to be the identity matrix, and if\nA\nis invertible, then.\nMatrix powers appear often in the context of [discrete dynamical systems](https://bliptext.com/articles/discrete-dynamical-system), where the matrix\nA\nexpresses a transition from a state vector\nx\nof some system to the next state\nAx\nof the system.\nThis is the standard interpretation of a [Markov chain](https://bliptext.com/articles/markov-chain), for example.\nThen A^2x is the state of the system after two time steps, and so forth: A^nx is the state of the system after\nn\ntime steps.\nThe matrix power A^n is the transition matrix between the state now and the state at a time\nn\nsteps in the future.\nSo computing matrix powers is equivalent to solving the evolution of the dynamical system.\nIn many cases, matrix powers can be expediently computed by using [eigenvalues and eigenvectors](https://bliptext.com/articles/eigenvalues-and-eigenvectors).\nApart from matrices, more general [linear operators](https://bliptext.com/articles/linear-operator) can also be exponentiated.\nAn example is the [derivative](https://bliptext.com/articles/derivative) operator of calculus, d/dx, which is a linear operator acting on functions f(x) to give a new function.\nThe\nn\nth power of the differentiation operator is the\nn\nth derivative:\nThese 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.\nThis is the starting point of the mathematical theory of [semigroups](https://bliptext.com/articles/c0-semigroup).\nJust as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics.\nExamples include approaches to solving the [heat equation](https://bliptext.com/articles/heat-equation), [Schrödinger equation](https://bliptext.com/articles/schr-dinger-equation), [wave equation](https://bliptext.com/articles/wave-equation), and other partial differential equations including a time evolution.\nThe special case of exponentiating the derivative operator to a non-integer power is called the [fractional derivative](https://bliptext.com/articles/fractional-derivative) which, together with the [fractional integral](https://bliptext.com/articles/fractional-integral), is one of the basic operations of the [fractional calculus](https://bliptext.com/articles/fractional-calculus).\n\n## Finite fields\nA [field](https://bliptext.com/articles/field-mathematics) is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is [associative](https://bliptext.com/articles/associative) and every nonzero element has a [multiplicative inverse](https://bliptext.com/articles/multiplicative-inverse).\nThis implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of\n0\n. Common examples are the field of [complex numbers](https://bliptext.com/articles/complex-number), the [real numbers](https://bliptext.com/articles/real-number) and the [rational numbers](https://bliptext.com/articles/rational-number), considered earlier in this article, which are all [infinite](https://bliptext.com/articles/infinite-set).\nA finite field is a field with a [finite number](https://bliptext.com/articles/finite-set) of elements.\nThis number of elements is either a [prime number](https://bliptext.com/articles/prime-number) or a [prime power](https://bliptext.com/articles/prime-power); that is, it has the form q=p^k, where p is a [prime number](https://bliptext.com/articles/prime-number), and k is a positive integer.\nFor every such q, there are fields with q elements.\nThe fields with q elements are all [isomorphic](https://bliptext.com/articles/isomorphic), which allows, in general, working as if there were only one field with q elements, denoted\nOne has\nfor every\nA [primitive element](https://bliptext.com/articles/primitive-element-finite-field) in \\mathbb F_q is an element g such that the set of the\nq − 1\nfirst powers of g (that is, ) equals the set of the nonzero elements of There are primitive elements in where \\varphi is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function).\nIn the [freshman's dream](https://bliptext.com/articles/freshman-s-dream) identity\nis true for the exponent p. As x^p=x in It follows that the map\nis [linear](https://bliptext.com/articles/linear-map) over and is a [field automorphism](https://bliptext.com/articles/field-automorphism), called the [Frobenius automorphism](https://bliptext.com/articles/frobenius-automorphism).\nIf q=p^k, the field \\mathbb F_q has k automorphisms, which are the k first powers (under [composition](https://bliptext.com/articles/function-composition)) of F. In other words, the [Galois group](https://bliptext.com/articles/galois-group) of \\mathbb F_q is [cyclic](https://bliptext.com/articles/cyclic-group) of order k, generated by the Frobenius automorphism.\nThe [Diffie–Hellman key exchange](https://bliptext.com/articles/diffie-hellman-key-exchange) is an application of exponentiation in finite fields that is widely used for [secure communications](https://bliptext.com/articles/secure-communication).\nIt uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the [discrete logarithm](https://bliptext.com/articles/discrete-logarithm), is computationally expensive.\nMore precisely, if g is a primitive element in then g^e can be efficiently computed with [exponentiation by squaring](https://bliptext.com/articles/exponentiation-by-squaring) for any e, even if q is large, while there is no known computationally practical algorithm that allows retrieving e from g^e if q is sufficiently large.\n\n# Powers of sets\nThe [Cartesian product](https://bliptext.com/articles/cartesian-product) of two [sets](https://bliptext.com/articles/set-mathematics) S and T is the set of the [ordered pairs](https://bliptext.com/articles/ordered-pair) (x,y) such that x\\in S and y\\in T. This operation is not properly [commutative](https://bliptext.com/articles/commutative) nor [associative](https://bliptext.com/articles/associative), but has these properties [up to](https://bliptext.com/articles/up-to) [canonical](https://bliptext.com/articles/canonical-map) [isomorphisms](https://bliptext.com/articles/isomorphism), that allow identifying, for example, (x,(y,z)), ((x,y),z), and (x,y,z).\nThis allows defining the nth power S^n of a set S as the set of all n-[tuples](https://bliptext.com/articles/tuple) of elements of S.\nWhen S is endowed with some structure, it is frequent that S^n is naturally endowed with a similar structure.\nIn this case, the term \"[direct product](https://bliptext.com/articles/direct-product)\" is generally used instead of \"Cartesian product\", and exponentiation denotes product structure.\nFor example \\R^n (where \\R denotes the real numbers) denotes the Cartesian product of n copies of \\R, as well as their direct product as [vector space](https://bliptext.com/articles/vector-space), [topological spaces](https://bliptext.com/articles/topological-space), [rings](https://bliptext.com/articles/ring-mathematics), etc.\n\n## Sets as exponents\nA n-tuple of elements of S can be considered as a [function](https://bliptext.com/articles/function-mathematics) from This generalizes to the following notation.\nGiven two sets S and T, the set of all functions from T to S is denoted S^T.\nThis exponential notation is justified by the following canonical isomorphisms (for the first one, see [Currying](https://bliptext.com/articles/currying)):\nwhere \\times denotes the Cartesian product, and \\sqcup the [disjoint union](https://bliptext.com/articles/disjoint-union).\nOne can use sets as exponents for other operations on sets, typically for [direct sums](https://bliptext.com/articles/direct-sum) of [abelian groups](https://bliptext.com/articles/abelian-group), [vector spaces](https://bliptext.com/articles/vector-space), or [modules](https://bliptext.com/articles/module-mathematics).\nFor distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses.\nFor example, \\R^\\N denotes the vector space of the [infinite sequences](https://bliptext.com/articles/infinite-sequence) of real numbers, and \\R^{(\\N)} the vector space of those sequences that have a finite number of nonzero elements.\nThe latter has a [basis](https://bliptext.com/articles/basis-linear-algebra) consisting of the sequences with exactly one nonzero element that equals\n1\n, while the [Hamel bases](https://bliptext.com/articles/hamel-basis) of the former cannot be explicitly described (because their existence involves [Zorn's lemma](https://bliptext.com/articles/zorn-s-lemma)).\nIn this context,\n2\ncan represents the set \\{0,1\\}.\nSo, 2^S denotes the [power set](https://bliptext.com/articles/power-set) of S, that is the set of the functions from S to \\{0,1\\}, which can be identified with the set of the [subsets](https://bliptext.com/articles/subset) of S, by mapping each function to the [inverse image](https://bliptext.com/articles/inverse-image) of\n1\n.\nThis fits in with the [exponentiation of cardinal numbers](https://bliptext.com/articles/cardinal-exponentiation), in the sense that\n=\n, where\nis the cardinality of\nX\n.\n\n## In category theory\nIn the [category of sets](https://bliptext.com/articles/category-of-sets), the [morphisms](https://bliptext.com/articles/morphism) between sets X and Y are the functions from X to Y. It results that the set of the functions from X to Y that is denoted Y^X in the preceding section can also be denoted \\hom(X,Y).\nThe isomorphism can be rewritten\nThis means the functor \"exponentiation to the power T\" is a [right adjoint](https://bliptext.com/articles/right-adjoint) to the functor \"direct product with T\".\nThis generalizes to the definition of [exponentiation in a category](https://bliptext.com/articles/exponential-category-theory) in which finite [direct products](https://bliptext.com/articles/direct-product) exist: in such a category, the functor X\\to X^T is, if it exists, a right adjoint to the functor A category is called a Cartesian closed category, if [direct products](https://bliptext.com/articles/direct-product) exist, and the functor has a right adjoint for every T.\n\n# Repeated exponentiation\nJust as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or [tetration](https://bliptext.com/articles/tetration).\nIterating tetration leads to another operation, and so on, a concept named [hyperoperation](https://bliptext.com/articles/hyperoperation).\nThis sequence of operations is expressed by the [Ackermann function](https://bliptext.com/articles/ackermann-function) and [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation).\nJust as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation.\nEvaluated at\n(3, 3)\n, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7,625,597,484,987 (\n=327 = 33 3 = 33\n) respectively.\n\n# Limits of powers\n[Zero to the power of zero](https://bliptext.com/articles/zero-to-the-power-of-zero) gives a number of examples of limits that are of the [indeterminate form](https://bliptext.com/articles/indeterminate-form) 00.\nThe limits in these examples exist, but have different values, showing that the two-variable function\nxy\nhas no limit at the point\n(0, 0)\n. One may consider at what points this function does have a limit.\nMore precisely, consider the function defined on.\nThen\nD\ncan be viewed as a subset of\nR2\n(that is, the set of all pairs\n(x, y)\nwith\nx\n,\ny\nbelonging to the [extended real number line](https://bliptext.com/articles/extended-real-number-line)\nR = [−∞, +∞]\n, endowed with the [product topology](https://bliptext.com/articles/product-topology)), which will contain the points at which the function\nf\nhas a limit.\nIn fact,\nf\nhas a limit at all [accumulation points](https://bliptext.com/articles/accumulation-point) of\nD\n, except for\n(0, 0)\n,\n(+∞, 0)\n,\n(1, +∞)\nand\n(1, −∞)\n. Accordingly, this allows one to define the powers\nxy\nby continuity whenever\n0 ≤ x ≤ +∞\n,\n−∞ ≤ y ≤ +∞\n, except for\n00\n,\n(+∞)0\n,\n1+∞\nand\n1−∞\n, which remain indeterminate forms.\nUnder this definition by continuity, we obtain:\nx+∞ = +∞\nand\nx−∞ = 0\n, when\n1 < x ≤ +∞\n.\nx+∞ = 0\nand\nx−∞ = +∞\n, when\n0 ≤ x < 1\n.\n0y = 0\nand\n(+∞)y = +∞\n, when\n0 < y ≤ +∞\n.\n0y = +∞\nand\n(+∞)y = 0\n, when\n−∞ ≤ y < 0\n.\nThese powers are obtained by taking limits of\nxy\nfor positive values of\nx\n. This method does not permit a definition of\nxy\nwhen\nx < 0\n, since pairs\n(x, y)\nwith\nx < 0\nare not accumulation points of\nD\n.\nOn the other hand, when\nn\nis an integer, the power\nxn\nis already meaningful for all values of\nx\n, including negative ones.\nThis may make the definition\n0n = +∞\nobtained above for negative\nn\nproblematic when\nn\nis odd, since in this case\nxn → +∞\nas\nx\ntends to\n0\nthrough positive values, but not negative ones.\n\n# Efficient computation with integer exponents\nComputing\nbn\nusing iterated multiplication requires\nn − 1\nmultiplication operations, but it can be computed more efficiently than that, as illustrated by the following example.\nTo compute\n2100\n, apply [Horner's rule](https://bliptext.com/articles/horner-s-rule) to the exponent 100 written in binary:\nThen compute the following terms in order, reading Horner's rule from right to left.\nThis series of steps only requires 8 multiplications instead of 99.\nIn general, the number of multiplication operations required to compute\nbn\ncan be reduced to by using [exponentiation by squaring](https://bliptext.com/articles/exponentiation-by-squaring), where \\sharp n denotes the number of\n1\ns in the [binary representation](https://bliptext.com/articles/binary-representation) of n. 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](https://bliptext.com/articles/addition-chain-exponentiation).\nFinding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for\nbn\nis a difficult problem, for which no efficient algorithms are currently known (see [Subset sum problem](https://bliptext.com/articles/subset-sum-problem)), but many reasonably efficient heuristic algorithms are available.\nHowever, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.\n\n# Iterated functions\n[Function composition](https://bliptext.com/articles/function-composition) is a [binary operation](https://bliptext.com/articles/binary-operation) that is defined on [functions](https://bliptext.com/articles/function-mathematics) such that the [co[domain](https://bliptext.com/articles/domain-of-a-function)](https://bliptext.com/articles/co[domain](https://bliptext.com/articles/domain-of-a-function)) of the function written on the right is included in the [domain](https://bliptext.com/articles/domain-of-a-function) of the function written on the left.\nIt is denoted g\\circ f, and defined as for every x in the domain of f.\nIf the domain of a function f equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the nth power of the function under composition, commonly called the nth iterate of the function.\nThus f^n denotes generally the nth iterate of f; for example, f^3(x) means f(f(f(x))).\nWhen a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the [pointwise multiplication](https://bliptext.com/articles/pointwise-multiplication), which induces another exponentiation.\nWhen using [functional notation](https://bliptext.com/articles/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.\nThus and When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example and For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the [trigonometric functions](https://bliptext.com/articles/trigonometric-functions).\nSo, \\sin^2 x and \\sin^2(x) both mean and not which, in any case, is rarely considered.\nHistorically, several variants of these notations were used by different authors.\nIn this context, the exponent -1 denotes always the [inverse function](https://bliptext.com/articles/inverse-function), if it exists.\nSo For the [multiplicative inverse](https://bliptext.com/articles/multiplicative-inverse) fractions are generally used as in\n\n# In programming languages\n[Programming languages](https://bliptext.com/articles/programming-language) generally express exponentiation either as an infix [operator](https://bliptext.com/articles/operator-computer-programming) or as a function application, as they do not support superscripts.\nThe most common operator symbol for exponentiation is the [caret](https://bliptext.com/articles/caret).\nThe [original version of ASCII](https://bliptext.com/articles/ascii) included an uparrow symbol, intended for exponentiation, but this was [replaced by the caret](https://bliptext.com/articles/caret) in 1967, so the caret became usual in programming languages.\nThe notations include:\nIn most programming languages with an infix exponentiation operator, it is [right-associative](https://bliptext.com/articles/operator-associativity), that is, is interpreted as.\nThis is because is equal to and thus not as useful.\nIn some languages, it is left-associative, notably in [Algol](https://bliptext.com/articles/algol), [MATLAB](https://bliptext.com/articles/matlab), and the [Microsoft Excel](https://bliptext.com/articles/microsoft-office-excel) formula language.\nOther programming languages use functional notation:\nStill others only provide exponentiation as part of standard [libraries](https://bliptext.com/articles/library-computing):\nIn some [statically typed](https://bliptext.com/articles/type-system) languages that prioritize [type safety](https://bliptext.com/articles/type-safety) such as [Rust](https://bliptext.com/articles/rust-programming-language), exponentiation is performed via a multitude of methods:","revisionCount":0},"hasMore":false,"nextCursor":null}