Exponentiation is a

complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

exponent is defined by means of the exponential function with complex argument (see the end of , above) as
:$b^z\; =\; e^,$
where $\backslash ln\; b$ denotes the

principal value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of the

multivalued function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s.
If $\backslash log\; z$ denotes one of the values of the multivalued logarithm (typically its principal value), the other values are $2ik\backslash pi\; +\backslash log\; z,$ where is any integer. Similarly, if $z^w$ is one value of the exponentiation, then the other values are given by
:$e^\; =\; z^we^,$
where is any integer.
Different values of give different values of $z^w$ unless is a

principal value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of this logarithm is $\backslash log\; z=\backslash ln\; \backslash rho+i\backslash theta,$ where $\backslash ln$ denotes the

The polar form of is $i=e^,$ and the values of $\backslash log\; i$ are thus $$\backslash log\; i=i\backslash left(\backslash frac\; \backslash pi\; 2\; +2k\backslash pi\backslash right).$$ It follows that $$i^i=e^=e^\; e^.$$So, all values of $i^i$ are real, the principal one being $$e^\; \backslash approx\; 0.2079.$$ *$(-2)^$

Similarly, the polar form of is $-2\; =\; 2e^.$ So, the above described method gives the values $$\backslash begin\; (-2)^\; \&=\; 2^3\; e^\; (\backslash cos(4\backslash ln\; 2\; +\; 3(\backslash pi\; +2k\backslash pi))\; +i\backslash sin(4\backslash ln\; 2\; +\; 3(\backslash pi+2k\backslash pi)))\backslash \backslash \; \&=-2^3\; e^(\backslash cos(4\backslash ln\; 2)\; +i\backslash sin(4\backslash ln\; 2)).\; \backslash end$$In this case, all the values have the same argument $4\backslash ln\; 2,$ and different absolute values. In both examples, all values of $z^w$ have the same argument. More generally, this is true if and only if the

multivalued function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

) are algebraic. If is

associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, but has these properties

^{0}. The limits in these examples exist, but have different values, showing that the two-variable function has no limit at the point . One may consider at what points this function does have a limit.
More precisely, consider the function defined on Then can be viewed as a subset of (that is, the set of all pairs with , belonging to the ^{0}, (+∞)^{0}, 1^{+∞} and 1^{−∞}, which remain indeterminate forms.
Under this definition by continuity, we obtain:
* and , when .
* and , when .
* and , when .
* and , when .
These powers are obtained by taking limits of for ''positive'' values of . This method does not permit a definition of when , since pairs with are not accumulation points of .
On the other hand, when is an integer, the power is already meaningful for all values of , including negative ones. This may make the definition obtained above for negative problematic when is odd, since in this case as tends to through positive values, but not negative ones.

^{''n''} using iterated multiplication requires multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2^{100}, apply Horner's rule to the exponent 100 written in binary:
:$100\; =\; 2^2\; +2^5\; +\; 2^6\; =\; 2^2(1+2^3(1+2))$.
Then compute the following terms in order, reading Horner's rule from right to left.
This series of steps only requires 8 multiplications instead of 99.
In general, the number of multiplication operations required to compute can be reduced to $\backslash sharp\; n\; +\backslash lfloor\; \backslash log\_\; n\backslash rfloor\; -1,$ by using

^{''y''} = ''y''^{''x''}
*

mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

operation, written as , involving two numbers, the ''base
Base or BASE may refer to:
Brands and enterprises
* Base (mobile telephony provider), a Belgian mobile telecommunications operator
*Base CRM
Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...

'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

, exponentiation corresponds to repeated multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

of the base: that is, is the product of multiplying bases:
:$$b^n\; =\; \backslash underbrace\_.$$
The exponent is usually shown as a superscript
Pro; the size of the subscript is about 62% of the original characters, dropped below the baseline by about 16%. The second typeface is Myriad
A myriad (from Ancient Greek
Ancient Greek includes the forms of the Greek language used in a ...

to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' raised to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th".
One has , and, for any positive integers and , one has . To extend this property to non-positive integer exponents, is defined to be , and (with a positive integer and not zero) is defined as . In particular, is equal to , the ''reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another poly ...

'' of .
The definition of exponentiation can be extended to allow any real or complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

.
Exponentiation is used extensively in many fields, including economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
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, biology
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, chemistry
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, physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, and computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

, with applications such as compound interest
Compound interest is the addition of interest
Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

, population growth
Population growth is the increase in the number of people in a population
Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size ...

, chemical reaction kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry
Physical chemistry is the study of macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with ...

, wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

behavior, and public-key cryptography
Public-key cryptography, or asymmetric cryptography, is a cryptographic system that uses pairs of keys
KEYS (1440 AM broadcasting, AM) is a radio station serving the Corpus Christi, Texas, Corpus Christi, Texas area with a talk radio, talk ...

.
History of the notation

The term ''power'' ( la, potentia, potestas, dignitas) is a mistranslation of theancient Greek
Ancient Greek includes the forms of the Greek language
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...

δύναμις (''dúnamis'', here: "amplification") used by the Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...

mathematician Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

for the square of a line, following Hippocrates of Chios
Hippocrates of Kos (; grc-gre, Ἱπποκράτης ὁ Κῷος, Hippokrátēs ho Kôios; ), also known as Hippocrates II, was a Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), ...

. In ''The Sand Reckoner
''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathemat ...

'', Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

discovered and proved the law of exponents, , necessary to manipulate powers of . In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), or al-Khwarizmi and formerly Latinisation of names, Latinized as ''Algorithmi'', was a Persians, Persian polymath who produced ...

used the terms مَال (''māl'', "possessions", "property") for a square
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"—and كَعْبَة ('''', "cube") for a cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, which later Islamic
Islam (; ar, اَلْإِسْلَامُ, al-’Islām, "submission

mathematicians represented in o God
Oh God may refer to:
* An exclamation; similar to "oh no", "oh yes", "oh my", "aw goodness", "ah gosh", "ah gawd"; see interjection
An interjection is a word or expression that occurs as an utterance on its own and expresses a spontaneous feeling ...

) is an Abrahamic religions, Abrahamic monotheistic religion teaching that Muhammad is a Muhammad in Islam, messenger of God.Peters, F. E. 2009. "Allāh." In , ed ...mathematical notation
Mathematical notation is a system of symbol
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or prag ...

as the letters '' mīm'' (m) and ''kāf
Kaf (also spelled kaph) is the eleventh letter of the Semitic abjads, including Phoenician Kāp , Hebrew
Hebrew (, , or ) is a Northwest Semitic languages, Northwest Semitic language of the Afroasiatic languages, Afroasiatic language fam ...

'' (k), respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.
In the late 16th century, Jost Bürgi
Jost Bürgi (also ''Joost, Jobst''; Latinized surname ''Burgius'' or ''Byrgius''; 28 February 1552 – 31 January 1632), active primarily at the courts in Kassel
Kassel (; in Germany, spelled Cassel until 1926) is a city on the Fulda River in ...

used Roman numerals for exponents.
Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus
Henricus Grammateus (also known as Henricus Scriptor, Heinrich Schreyber or Heinrich Schreiber; 1495 – 1525 or 1526) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from ...

and Michael Stifel
Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the ...

in the 16th century. The word ''exponent'' was coined in 1544 by Michael Stifel. Samuel Jeake introduced the term ''indices'' in 1696. In the 16th century, Robert Recorde
Robert Recorde (c. 1512 – 1558) was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign
The plus and minus signs, and , are mathematical symbols used to represent the notions ...

used the terms square, cube, zenzizenzic (fourth power
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic
Zenzizenzizenzic is an obsolete form of mathematical notation representing the eighth power of a number (that is, the zenzizenzizenzic of ''x'' is ''x''8), dating from a time when powers were written out in words rather than as superscript numbers. ...

(eighth). ''Biquadrate'' has been used to refer to the fourth power as well.
Early in the 17th century, the first form of our modern exponential notation was introduced by René Descartes
René Descartes ( or ; ; Latinized
Latinisation or Latinization can refer to:
* Latinisation of names, the practice of rendering a non-Latin name in a Latin style
* Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

in his text titled ''La Géométrie
''La Géométrie'' was published
Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the term refers to the distribution of printed works ...

''; there, the notation is introduced in Book I.
Some mathematicians (such as Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, for example, as .
Another historical synonym, involution, is now rare and should not be confused with its more common meaning.
In 1748, Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

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 notalgebraic functionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...s, since in those the exponents must be constant."

Terminology

The expression is called "thesquare
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

of ''b''" or "''b'' squared", because the area of a square with side-length is .
Similarly, the expression is called "the cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

of ''b''" or "''b'' cubed", because the volume of a cube with side-length is .
When it is a positive integer
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the multiplication, because the exponent is . Here, is the ''5th power of 3'', or ''3 raised to the 5th power''.
The word "raised" is usually omitted, and sometimes "power" as well, so can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation can be expressed as "''b'' to the power of ''n''", "''b'' to the ''n''th power", "''b'' to the ''n''th", or most briefly as "''b'' to the ''n''".
A formula with nested exponentiation, such as (which means and not ), is called a tower of powers, or simply a tower.
Integer exponents

The exponentiation operation with integer exponents may be defined directly from elementaryarithmetic operation
Arithmetic (from the Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

s.
Positive exponents

The definition of the exponentiation as an iterated multiplication can be formalized by usinginduction
Induction may refer to:
Philosophy
* Inductive reasoning, in logic, inferences from particular cases to the general case
Biology and chemistry
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induction period, the t ...

, and this definition can be used as soon one has an associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

multiplication:
The base case is
:$b^1\; =\; b$
and the recurrence is
:$b^\; =\; b^n\; \backslash cdot\; b.$
The associativity of multiplication implies that for any positive integers and ,
:$b^\; =\; b^m\; \backslash cdot\; b^n,$
and
:$(b^m)^n=b^.$
Zero exponent

By definition, any nonzero number raised to the power is : :$b^0=1.$ This definition is the only possible that allows extending the formula :$b^=b^m\backslash cdot\; b^n$ to zero exponents. It may be used in everyalgebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

with a multiplication that has an identity
Identity may refer to:
Social sciences
* Identity (social science), personhood or group affiliation in psychology and sociology
Group expression and affiliation
* Cultural identity, a person's self-affiliation (or categorization by others ...

.
Intuitionally, $b^0$ may be interpreted as the empty product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of copies of . So, the equality $b^0=1$ is a special case of the general convention for the empty product.
The case of is more complicated. In contexts where only integer powers are considered, the value is generally assigned to $0^0,$ but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.
Negative exponents

Exponentiation with negative exponents is defined by the following identity, which holds for any integer and nonzero : :$b^\; =\; \backslash frac.$ Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity ($\backslash infty$). This definition of exponentiation with negative exponents is the only one that allows extending the identity $b^=b^m\backslash cdot\; b^n$ to negative exponents (consider the case $m=-n$). The same definition applies toinvertible element
In the branch of abstract algebra known as ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...

s in a multiplicative monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

, that is, an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, with an associative multiplication and a multiplicative identity
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...

denoted (for example, the square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a given dimension). In particular, in such a structure, the inverse of an invertible element
In the branch of abstract algebra known as ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...

is standardly denoted $x^.$
Identities and properties

The following identities, often called , hold for all integer exponents, provided that the base is non-zero: :$\backslash begin\; b^\; \&=\; b^m\; \backslash cdot\; b^n\; \backslash \backslash \; \backslash left(b^m\backslash right)^n\; \&=\; b^\; \backslash \backslash \; (b\; \backslash cdot\; c)^n\; \&=\; b^n\; \backslash cdot\; c^n\; \backslash end$ Unlike addition and multiplication, exponentiation is notcommutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. For example, . Also unlike addition and multiplication, exponentiation is not associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. For example, , whereas . Without parentheses, the conventional order of operations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

for serial exponentiation in superscript notation is top-down (or ''right''-associative), not bottom-up (or ''left''-associative). That is,
:$b^\; =\; b^,$
which, in general, is different from
:$\backslash left(b^p\backslash right)^q\; =\; b^\; .$
Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by thebinomial formula
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of exponentiation, powers of a binomial (polynomial), binomial. According to the theorem, it is possible to expand the polynomial into a summati ...

:$(a+b)^n=\backslash sum\_^n\; \backslash binoma^ib^=\backslash sum\_^n\; \backslash fraca^ib^.$
However, this formula is true only if the summands commute (i.e. that ), which is implied if they belong to a structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

that is commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. Otherwise, if and are, say, square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the same size, this formula cannot be used. It follows that in computer algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, many algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software
Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data.
It is a type of applica ...

s use a different notation (sometimes instead of ) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.
Combinatorial interpretation

For nonnegative integers and , the value of is the number offunctions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

from a set of elements to a set of elements (see cardinal exponentiation). Such functions can be represented as -tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s from an -element set (or as -letter words from an -letter alphabet). Some examples for particular values of and are given in the following table:
:
Particular bases

Powers of ten

In the base ten (decimal
The decimal numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ''communicare'', meaning "to share") is t ...

) number system, integer powers of are written as the digit followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, and .
Exponentiation with base is used in scientific notation
Scientific notation is a way of expressing numbers
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or coul ...

to denote large or small numbers. For instance, (the speed of light
The speed of light in vacuum
A vacuum is a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...

in vacuum, in metres per second
The metre per second is an SI derived unit
SI derived units are units of measurement derived from the
seven SI base unit, base units specified by the International System of Units (SI). They are either dimensionless quantity, dimensionless or ...

) can be written as and then approximated as .
SI prefix
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...

es based on powers of are also used to describe small or large quantities. For example, the prefix kilo
KILO (94.3 FM, 94.3 KILO) is a radio station broadcasting in Colorado Springs
The City of Colorado Springs is the List of cities and towns in Colorado#Home rule municipality, Home Rule Municipality that is the county seat and the List of ci ...

means , so a kilometre is .
Powers of two

The first negative powers of are commonly used, and have special names, e.g.: ''half
One half is the irreducible fraction resulting from dividing 1 (number), one by 2 (number), two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or "halving"; con ...

'' and ''''.
Powers of appear in set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

, since a set with members has a power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, the set of all of its subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, which has members.
Integer powers of are important in computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

. The positive integer powers give the number of possible values for an -bit
The bit is a basic unit of information in computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm
of an algorithm (Euclid's algo ...

integer binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons
The electr ...

; for example, a byte
The byte is a unit of digital information that most commonly consists of eight bit
The bit is a basic unit of information in computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It ...

may take different values. The binary number system
In mathematics and digital electronics
Digital electronics is a field of electronics
Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter ...

expresses any number as a sum of powers of , and denotes it as a sequence of and , separated by a binary pointIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, where indicates a power of that appears in the sum; the exponent is determined by the place of this : the nonnegative exponents are the rank of the on the left of the point (starting from ), and the negative exponents are determined by the rank on the right of the point.
Powers of one

The powers of one are all one: . The first power of a number is the number itself: $n^1=n.$Powers of zero

If the exponent is positive (), the th power of zero is zero: . If the exponent is negative (), the th power of zero is undefined, because it must equal $1/0^$ with , and this would be $1/0$ according to above. The expression is either defined as 1, or it is left undefined.Powers of negative one

If is an even integer, then . If is an odd integer, then . Because of this, powers of are useful for expressing alternatingsequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s. For a similar discussion of powers of the complex number , see .
Large exponents

Thelimit of a sequence
As the positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. ...

of powers of a number greater than one diverges; in other words, the sequence grows without bound:
: as when
This can be read as "''b'' to the power of ''n'' tends to +∞ as ''n'' tends to infinity when ''b'' is greater than one".
Powers of a number with absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

less than one tend to zero:
: as when
Any power of one is always one:
: for all if
Powers of alternate between and as alternates between even and odd, and thus do not tend to any limit as grows.
If , , alternates between larger and larger positive and negative numbers as alternates between even and odd, and thus does not tend to any limit as grows.
If the exponentiated number varies while tending to as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
: as
See ' below.
Other limits, in particular those of expressions that take on an indeterminate formIn calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...

, are described in below.
Power functions

Real functions of the form $f(x)\; =\; cx^n$, where $c\; \backslash ne\; 0$, are sometimes called power functions. When $n$ is aninteger
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

and $n\; \backslash ge\; 1$, two primary families exist: for $n$ even, and for $n$ odd. In general for $c\; >\; 0$, when $n$ is even $f(x)\; =\; cx^n$ will tend towards positive infinity
Infinity is that which is boundless, endless, or larger than any number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything ...

with increasing $x$, and also towards positive infinity 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. Functions with this kind of symmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

are called even functions.
When $n$ is odd, $f(x)$'s asymptotic
250px, A curve intersecting an asymptote infinitely many times.
In analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc ...

behavior reverses from positive $x$ to negative $x$. For $c\; >\; 0$, $f(x)\; =\; cx^n$ will also tend towards positive infinity
Infinity is that which is boundless, endless, or larger than any number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything ...

with increasing $x$, but towards negative infinity 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 $n=1$. Functions with this kind of symmetry are called odd functions.
For $c\; <\; 0$, the opposite asymptotic behavior is true in each case.
Table of powers of decimal digits

Rational exponents

If is a nonnegativereal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

, and is a positive integer, $x^\backslash frac\; 1n$ or $\backslash sqrt;\; href="/html/ALL/s/.html"\; ;"title="">$ denotes the unique positive real th root of , that is, the unique positive real number such that $y^n=x.$
If is a positive real number, and $\backslash frac\; pq$ is a rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

, with and integers, then $x^\backslash frac\; pq$ is defined as
:$x^\backslash frac\; pq=\; \backslash left(x^p\backslash right)^\backslash frac\; 1q=(x^\backslash frac\; 1q)^p.$
The equality on the right may be derived by setting $y=x^\backslash frac\; 1q,$ and writing $(x^\backslash frac\; 1q)^p=y^p=\backslash left((y^p)^q\backslash right)^\backslash frac\; 1q=\backslash left((y^q)^p\backslash right)^\backslash frac\; 1q=(x^p)^\backslash frac\; 1q.$
If is a positive rational number, $0^r=0,$ by definition.
All these definitions are required for extending the identity $(x^r)^s\; =\; x^$ to rational exponents.
On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real th root, which is negative if is odd, and no real root if is even. In the latter case, whichever complex th root one chooses for $x^\backslash frac\; 1n,$ the identity $(x^a)^b=x^$ cannot be satisfied. For example,
:$\backslash left((-1)^2\backslash right)^\backslash frac\; 12\; =\; 1^\backslash frac\; 12=\; 1\backslash neq\; (-1)^\; =(-1)^1=-1.$
See and for details on the way these problems may be handled.
Real exponents

For 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 thelogarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

of the base and the exponential function
The exponential function is a mathematical function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of ...

(, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

exponents.
On 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 (see ). One may choose one of these values, called the principal value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, but there is no choice of the principal value for which the identity
:$\backslash left(b^r\backslash right)^s\; =\; b^$
is true; see . Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
Limits of rational exponents

Since anyirrational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

can be expressed as the limit of a sequence
As the positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. ...

of rational numbers, exponentiation of a positive real number with an arbitrary real exponent can be defined by continuity with the rule
:$b^x\; =\; \backslash lim\_\; b^r\; \backslash quad\; (b\; \backslash in\; \backslash mathbb^+,\backslash ,\; x\; \backslash in\; \backslash mathbb),$
where the limit is taken over rational values of only. This limit exists for every positive and every real .
For example, if , the non-terminating decimal representation and the monotonicity
Figure 3. A function that is not monotonic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain $b^\backslash pi:$
:$\backslash left;\; href="/html/ALL/s/^3,\_b^4\backslash right.html"\; ;"title="^3,\; b^4\backslash right">^3,\; b^4\backslash right$
So, the upper bounds and the lower bounds of the intervals form two sequences
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

that have the same limit, denoted $b^\backslash pi.$
This defines $b^x$ for every positive and real as a continuous function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of and . See also Well-defined expression.
The exponential function

The ''exponential function'' is often defined as $x\backslash mapsto\; e^x,$ where $e\backslash approx\; 2.718$ isEuler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of a logarithm, base of the natural logarithm. It is the Limit of a sequence, limit of ...

. For avoiding circular reasoning
Circular reasoning ( la, circulus in probando, "circle in proving"; also known as circular logic) is a logical fallacy in which the reasoner begins with what they are trying to end with. The components of a circular argument are often logically ...

, this definition cannot be used here. So, a definition of the exponential function, denoted $\backslash exp(x),$ and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has
:$\backslash exp(x)=e^x.$
There are many equivalent ways to define the exponential function, one of them being
:$\backslash exp(x)\; =\; \backslash lim\_\; \backslash left(1\; +\; \backslash frac\backslash right)^n.$
One has $\backslash exp(0)=1,$ and the ''exponential identity'' $\backslash exp(x+y)=\backslash exp(x)\backslash exp(y)$ holds as well, since
:$\backslash exp(x)\backslash exp(y)\; =\; \backslash lim\_\; \backslash left(1\; +\; \backslash frac\backslash right)^n\backslash left(1\; +\; \backslash frac\backslash right)^n\; =\; \backslash lim\_\; \backslash left(1\; +\; \backslash frac\; +\; \backslash frac\backslash right)^n,$
and the second-order term $\backslash frac$ does not affect the limit, yielding $\backslash exp(x)\backslash exp(y)\; =\; \backslash exp(x+y)$.
Euler's number can be defined as $e=\backslash exp(1)$. It follows from the preceding equations that $\backslash exp(x)=e^x$ when is an integer (this results from the repeated-multiplication definition of the exponentiation). If is real, $\backslash exp(x)=e^x$ results from the definitions given in preceding sections, by using the exponential identity if is rational, and the continuity of the exponential function otherwise.
The limit that defines the exponential function converges for every complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

value of , and therefore it can be used to extend the definition of $\backslash exp(z)$, and thus $e^z,$ from the real numbers to any complex argument . This extended exponential function still satifies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.
Powers via logarithms

The definition of as the exponential function allows defining for every positive real numbers , in terms of exponential andlogarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

function. Specifically, the fact that the natural logarithm
The natural logarithm of a number is its logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

is the inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when add ...

of the exponential function means that one has
: $b\; =\; \backslash exp(\backslash ln\; b)=e^$
for every . For preserving the identity $(e^x)^y=e^,$ one must have
:$b^x=\backslash left(e^\; \backslash right)^x\; =\; e^$
So, $e^$ can be used as an alternative definition of for any positive real . This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.
Complex exponents with a positive real base

If is a positive real number, exponentiation with base andnatural logarithm
The natural logarithm of a number is its logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

of .
This satisfies the identity
:$b^\; =\; b^z\; b^t,$
In general,
$\backslash left(b^z\backslash right)^t$ is not defined, since is not a real number. If a meaning is given to the exponentiation of a complex number (see , below), one has, in general,
:$\backslash left(b^z\backslash right)^t\; \backslash ne\; b^,$
unless is real or is integer.
Euler's formula
Euler's formula, named after Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) incl ...

,
:$e^\; =\; \backslash cos\; y\; +\; i\; \backslash sin\; y,$
allows expressing the polar form
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of $b^z$ in terms of the real and imaginary parts
In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...

of , namely
:$b^=\; b^x(\backslash cos(y\backslash ln\; b)+i\backslash sin(y\backslash ln\; b)),$
where the absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the trigonometric
Trigonometry (from Ancient Greek, Greek ''wikt:τρίγωνον, trigōnon'', "triangle" and ''wikt:μέτρον, metron'', "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The fiel ...

factor is one. This results from
:$b^=b^x\; b^=b^x\; e^\; =b^x(\backslash cos(y\backslash ln\; b)+i\backslash sin(y\backslash ln\; b)).$
Non-integer powers of complex numbers

In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of th roots, that is, of exponents $1/n,$ where is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to th roots, this case deserves to be considered first, since it does not need to usecomplex logarithm
of the color is used to show the ''arg
Arg or ARG may refer to:
Places
*''Arg'' () means "citadel" in Persian, and may refer to:
**Arg, Iran, a village in Fars Province, Iran
**Arg (Kabul), presidential palace in Kabul, Afghanistan
**Arg, South ...

s, and is therefore easier to understand.
th roots of a complex number

Every nonzero complex number may be written inpolar form
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

as
:$z=\backslash rho\; e^=r(cos\; \backslash theta\; +i\; \backslash sin\; \backslash theta),$
where $\backslash rho$ is the absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of , and $\backslash theta$ is its argument
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

. The argument is defined up to Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

an integer multiple of ; this means that, if $\backslash theta$ is the argument of a complex number, then $\backslash theta\; +2k\backslash pi$ is also an argument of the same complex number.
The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an th root of a complex number can be obtained by taking the th root of the absolute value and dividing its argument by :
:$\backslash left(\backslash rho\; e^\backslash right)^\backslash frac\; 1n=\backslash sqrt;\; href="/html/ALL/s/.html"\; ;"title="">$
If $2i\backslash pi$ is added to $\backslash theta,$ the complex number in not changed, but this adds $2i\backslash pi/n$ to the argument of the th root, and provides a new th root. This can be done times, and provides the th roots of the complex number.
It is usual to choose one of the th root as the principal root. The common choice is to choose the th root for which $-\backslash pi<\backslash theta\backslash le\; \backslash pi,$ that is, the th root that has the largest real part, and, if they are two, the one with positive imaginary part. This makes the principal th root a continuous function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

in the whole complex plane, except for negative real values of the radicand
In mathematics, an ''n''th root of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root of degree 2 is called a ...

. This function equals the usual th root for positive real radicands. For negative real radicands, and odd exponents, the principal th root is not real, although the usual th root is real. Analytic continuation
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic can also have the following meanings:
Natural sciences Chemistry
* ...

shows that the principal th root is the unique complex differentiable
A rectangular grid (top) and its image under a conformal map ''f'' (bottom).
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

function that extends the usual th root to the complex plane without the nonpositive real numbers.
If the complex number is moved around zero by increasing its argument, after an increment of $2\backslash pi,$ the complex number comes back to its initial position, and its th roots are (they are multiplied by e^). This shows that it is not possible to define a th root function that is not continuous in the whole complex plane.
Roots of unity

The th roots of unity are the complex numbers such that , where is a positive integer. They arise in various areas of mathematics, such as indiscrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...

or algebraic solutions of algebraic equations (Lagrange resolvent
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a ratio ...

).
The th roots of unity are the first powers of $\backslash omega\; =e^\backslash frac$, that is $1=\backslash omega^0=\backslash omega^n,\; \backslash omega=\backslash omega^1,\; \backslash omega^2,\; \backslash omega^.$ The th roots of unity that have this generating property are called ''primitive th roots of unity''; they have the form $\backslash omega^k=e^\backslash frac,$ with coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

with . The unique primitive square root of unity is $-1;$ the primitive fourth roots of unity are $i$ and $-i.$
The th roots of unity allow expressing all th roots of a complex number as the products of a given th roots of with a th root of unity.
Geometrically, the th roots of unity lie on the unit circle
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of the complex plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

at the vertices of a regular -gon with one vertex on the real number 1.
As the number $e^\backslash frac$ is the primitive th root of unity with the smallest positive argument
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

, it is called the ''principal primitive th root of unity'', sometimes shortened as ''principal th root of unity'', although this terminology can be confused with the principal value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of $1^$ which is 1.
Complex exponentiation

Defining 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. So, either aprincipal value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is defined, which is not continuous for the values of that are real and nonpositive, or z^w is defined as a multivalued function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
In all cases, the complex logarithm
of the color is used to show the ''arg
Arg or ARG may refer to:
Places
*''Arg'' () means "citadel" in Persian, and may refer to:
**Arg, Iran, a village in Fars Province, Iran
**Arg (Kabul), presidential palace in Kabul, Afghanistan
**Arg, South ...

is used to define complex exponentiation as
:$z^w=e^,$
where $\backslash log\; z$ is the variant of the complex logarithm that is used, which is, a function or a multivalued function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

such that
:$e^=z$
for every in its domain of definition.
Principal value

Thecomplex logarithm
of the color is used to show the ''arg
Arg or ARG may refer to:
Places
*''Arg'' () means "citadel" in Persian, and may refer to:
**Arg, Iran, a village in Fars Province, Iran
**Arg (Kabul), presidential palace in Kabul, Afghanistan
**Arg, South ...

is the unique function, commonly denoted $\backslash log,$ such that, for every nonzero complex number ,
:$e^=z,$
and the imaginary part
In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...

of satisfies
:$-\backslash pi\; <\backslash mathrm\; \backslash le\; \backslash pi.$
The principal value of the complex logarithm is not defined for $z=0,$ it is discontinuous
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all function (mathematics), functions are continuous. If a function is not continuous at a point in its domain of a function, domain, one says t ...

at negative real values of , and it is holomorphic
Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom).
In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...

(that is, complex differentiable) elsewhere. If is real and positive, the principal value of the complex logarithm is the natural logarithm: $\backslash log\; z=\backslash ln\; z.$
The principal value of $z^w$ is defined as
$z^w=e^,$
where $\backslash log\; z$ is the principal value of the logarithm.
The function $(z,w)\backslash to\; z^w$ is holomorphic except in the neighbourhood of the points where is real and nonpositive.
If is real and positive, the principal value of $z^w$ equals its usual value defined above. If $w=1/n,$ where is an integer, this principal value is the same as the one defined above.
Multivalued function

In some contexts, there is a problem with the discontinuity of the principal values of $\backslash log\; z$ and $z^w$ at the negative real values of . In this case, it is useful to consider these functions asrational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

, that is, there is an integer such that is an integer. This results from the of the exponential function, more specifically, that $e^a=e^b$ if and only if $a-b$ is an integer multiple of $2\backslash pi\; i.$
If $w=\backslash frac\; mn$ is a rational number with and coprime integers
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

with $n>0,$ then $z^w$ has exactly values. In the case $m=1,$ these values are the same as those described in § th roots of a complex number. If is an integer, there is only one value that agrees with that of .
The multivalued exponentiation is holomorphic for $z\backslash ne\; 0,$ in the sense that its graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

consists of several sheets that define each a holomorphic function in the neighborhood of every point. If varies continuously along a circle around , then, after a turn, the value of $z^w$ has changed of sheet.
Computation

The ''canonical form'' $x+iy$ of $z^w$ can be computed from the canonical form of and . Although this can be described by a single formula, it is clearer to split the computation in several steps. *''Polar form
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of ''. If $z=a+ib$ is the canonical form of ( and being real), then its polar form is $$z=\backslash rho\; e^=\; \backslash rho\; (\backslash cos\backslash theta\; +\; i\; \backslash sin\backslash theta),$$ where $\backslash rho=\backslash sqrt$ and $\backslash theta=\backslash operatorname(a,b)$ (see atan2
The function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automati ...

for the definition of this function).
*''Logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

of ''. The natural logarithm
The natural logarithm of a number is its logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

. The other values of the logarithm are obtained by adding $2ik\backslash pi$ for any integer .
*''Canonical form of $w\backslash log\; z.$'' If $w=c+di$ with and real, the values of $w\backslash log\; z$ are $$w\backslash log\; z\; =\; (c\backslash ln\; \backslash rho\; -\; d\backslash theta-2dk\backslash pi)\; +i\; (d\backslash ln\; \backslash rho\; +\; c\backslash theta+2ck\backslash pi),$$ the principal value corresponding to $k=0.$
*''Final result.'' Using the identities $e^=e^xe^y$ and $e^\; =x^y,$ one gets $$z^w=\backslash rho^c\; e^\; \backslash left(\backslash cos\; (d\backslash ln\; \backslash rho\; +\; c\backslash theta+2ck\backslash pi)\; +i\backslash sin(d\backslash ln\; \backslash rho\; +\; c\backslash theta+2ck\backslash pi)\backslash right),$$ with $k=0$ for the principal value.
=Examples

= * $i^i$The polar form of is $i=e^,$ and the values of $\backslash log\; i$ are thus $$\backslash log\; i=i\backslash left(\backslash frac\; \backslash pi\; 2\; +2k\backslash pi\backslash right).$$ It follows that $$i^i=e^=e^\; e^.$$So, all values of $i^i$ are real, the principal one being $$e^\; \backslash approx\; 0.2079.$$ *$(-2)^$

Similarly, the polar form of is $-2\; =\; 2e^.$ So, the above described method gives the values $$\backslash begin\; (-2)^\; \&=\; 2^3\; e^\; (\backslash cos(4\backslash ln\; 2\; +\; 3(\backslash pi\; +2k\backslash pi))\; +i\backslash sin(4\backslash ln\; 2\; +\; 3(\backslash pi+2k\backslash pi)))\backslash \backslash \; \&=-2^3\; e^(\backslash cos(4\backslash ln\; 2)\; +i\backslash sin(4\backslash ln\; 2)).\; \backslash end$$In this case, all the values have the same argument $4\backslash ln\; 2,$ and different absolute values. In both examples, all values of $z^w$ have the same argument. More generally, this is true if and only if the

real part
In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...

of is an integer.
Failure of power and logarithm identities

Some 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''. For example:Irrationality and transcendence

If is a positive realalgebraic number
An algebraic number is any complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...

, and is a rational number, then is an algebraic number. This results from the theory of algebraic extension
In abstract algebra, a field extension ''L''/''K'' is called algebraic if every element of ''L'' is algebraic over ''K'', i.e. if every element of ''L'' is a root
In vascular plants, the roots are the plant organ, organs of a plant that are ...

s. This remains true if is any algebraic number, in which case, all values of (as a irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. Th ...

(that is, ''not rational''), and both and are algebraic, Gelfond–Schneider theorem asserts that all values of are transcendental (that is, not algebraic), except if equals or .
In other words, if is irrational and $b\backslash not\backslash in\; \backslash ,$ then at least one of , and is transcendental.
Integer powers in algebra

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to anyassociative operation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

denoted as a multiplication.More generally, power associativityIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

is sufficient for the definition. The definition of $x^0$ requires further the existence of a multiplicative identity
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...

.
An algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

. In such a monoid, exponentiation of an element is defined inductively by
* $x^0\; =\; 1,$
* $x^\; =x\; x^n$ for every nonnegative integer .
If is a negative integer, $x^n$ is defined only if has a multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

. In this case, the inverse of is denoted $x^,$ and $x^n$ is defined as $\backslash left(x^\backslash right)^.$
Exponentiation with integer exponents obeys the following laws, for and in the algebraic structure, and and integers:
:$\backslash begin\; x^0\&=1\backslash \backslash \; x^\&=x^m\; x^n\backslash \backslash \; (x^m)^n\&=x^\backslash \backslash \; (xy)^n\&=x^n\; y^n\; \backslash quad\; \backslash text\; xy=yx,\; \backslash text\; \backslash end$
These definitions are widely used in many areas of mathematics, notably for groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

, rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

, fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...

, square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(which form a ring). They apply also to functions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

from a set to itself, which form a monoid under function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

. This includes, as specific instances, geometric transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, and endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...

s of any mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if is a real function
In mathematical analysis, and applications in geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...

whose valued can be multiplied, $f^n$ denotes the exponentiation with respect of multiplication, and $f^$ may denote exponentiation with respect of function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

. That is,
:$(f^n)(x)=(f(x))^n=f(x)\; \backslash ,f(x)\; \backslash cdots\; f(x),$
and
:$(f^)(x)=f(f(\backslash cdots\; f(f(x))\backslash cdots)).$
Commonly, $(f^n)(x)$ is denoted $f(x)^n,$ while $(f^)(x)$ is denoted $f^n(x).$
In a group

Amultiplicative group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a set with as associative operation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

denoted as multiplication, that has an identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

, and such that every element has an inverse.
So, if is a group, $x^n$ is defined for every $x\backslash in\; G$ and every integer .
The set of all powers of an element of a group form a subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

. A group (or subgroup) that consists of all powers of a specific element is the cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

generated by . If all the powers of are distinct, the group is isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to the additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structure ...

$\backslash Z$ of the integers. Otherwise, the cyclic group is finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

(it has a finite number of elements), and its number of elements is the order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

of . If the order of is , then $x^n=x^0=1,$ and the cyclic group generated by consists of the first powers of (starting indifferently from the exponent or ).
Order of elements play a fundamental role in group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the ''order'' of the group). The possible orders of group elements are important in the study of the structure of a group (see Sylow theorems
Peter Ludwig Mejdell Sylow () (12 December 1832 – 7 September 1918) was a Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic ...

), and in the classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

.
Superscript notation is also used for conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the change ...

; that is, , where ''g'' and ''h'' are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely $(g^h)^k=g^$ and $(gh)^k=g^kh^k.$
In a ring

In a ring, it may occur that some nonzero elements satisfy $x^n=0$ for some integer . Such an element is said to benilpotent
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. In a commutative ring
In ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...

, the nilpotent elements form an ideal
Ideal may refer to:
Philosophy
* Ideal (ethics)
An ideal is a principle
A principle is a proposition or value that is a guide for behavior or evaluation. In law
Law is a system
A system is a group of Interaction, interacting ...

, called the nilradical of the ring.
If the nilradical is reduced to the zero ideal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(that is, if $x\backslash neq\; 0$ implies $x^n\backslash neq\; 0$ for every positive integer ), the commutative ring is said reduced. Reduced rings important in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, since the coordinate ring
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

of an affine algebraic set
Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to:
*Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine combination, a certai ...

is always a reduced ring.
More generally, given an ideal in a commutative ring , the set of the elements of that have a power in is an ideal, called the radical
Radical may refer to:
Arts and entertainment Music
*Radical (mixtape), ''Radical'' (mixtape), by Odd Future, 2010
*Radical (Smack album), ''Radical'' (Smack album), 1988
*"Radicals", a song by Tyler, The Creator from the 2011 album ''Goblin (album ...

of . The nilradical is the radical of the zero ideal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. A radical ideal is an ideal that equals its own radical. In a polynomial ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$k;\; href="/html/ALL/s/\_1,\_\backslash ldots,\_x\_n.html"\; ;"title="\_1,\; \backslash ldots,\; x\_n">\_1,\; \backslash ldots,\; x\_n$field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, 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
Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''Satz
' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

).
Matrices and linear operators

If ''A'' is a square matrix, then the product of ''A'' with itself ''n'' times is called the matrix power. Also $A^0$ is defined to be the identity matrix, and if ''A'' is invertible, then $A^\; =\; \backslash left(A^\backslash right)^n$. Matrix powers appear often in the context of discrete dynamical systems, where the matrix ''A'' expresses a transition from a state vector ''x'' of some system to the next state ''Ax'' of the system. This is the standard interpretation of aMarkov chain
A Markov chain or Markov process is a stochastic model
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory tr ...

, for example. Then $A^2x$ is the state of the system after two time steps, and so forth: $A^nx$ is the state of the system after ''n'' time steps. The matrix power $A^n$ is the transition matrix between the state now and the state at a time ''n'' steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to it ...

.
Apart from matrices, more general linear operator
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s can also be exponentiated. An example is the derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

operator of calculus, $d/dx$, which is a linear operator acting on functions $f(x)$ to give a new function $(d/dx)f(x)\; =\; f\text{'}(x)$. The ''n''-th power of the differentiation operator is the ''n''-th derivative:
:$\backslash left(\backslash frac\backslash right)^nf(x)\; =\; \backslash fracf(x)\; =\; f^(x).$
These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...

. Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, Schrödinger equation
The Schrödinger equation is a linear
Linearity is the property of a mathematical relationship (''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a ma ...

, wave equation
The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and .
Historically, the problem of a such as that of a was studied by , , , and ...

, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the derivative, differentiation operator
:D f(x) = \frac f(x)\,,
and of the inte ...

which, together with the fractional integral, is one of the basic operations of the fractional calculus
Fractional calculus is a branch of mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), mea ...

.
Finite fields

Afield
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and every nonzero element has a multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of . Common examples are the complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s and their subfields, the rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s and the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s, which have been considered earlier in this article, and are all infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (band), a South Korean boy band
*''Infinite'' (EP), debut EP of American musi ...

.
A ''finite field'' is a field with a finite number of elements. This number of elements is either a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

or a prime power
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

; that is, it has the form $q=p^k,$ where is a prime number, and is a positive integer. For every such , there are fields with elements. The fields with elements are all isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, which allows, in general, working as if there were only one field with elements, denoted $\backslash mathbb\; F\_q.$
One has
:$x^q=x$
for every $x\backslash in\; \backslash mathbb\; F\_q.$
A primitive element in $\backslash mathbb\; F\_q$ is an element such the set of the first powers of (that is, $\backslash $) equals the set of the nonzero elements of $\backslash mathbb\; F\_q.$ There are $\backslash varphi\; (p-1)$ primitive elements in $\backslash mathbb\; F\_q,$ where $\backslash varphi$ is Euler's totient function
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...

.
In $\backslash mathbb\; F\_q,$ the Freshman's dream
The freshman's dream is a name sometimes given to the erroneous equation (''x'' + ''y'')''n'' = ''x'n'' + ''y'n'', where ''n'' is a real number (usually a positive integer greater than 1). Beginning students comm ...

identity
:$(x+y)^p\; =\; x^p+y^p$
is true for the exponent . As $x^p=x$ in $\backslash mathbb\; F\_q,$ It follows that the map
:$\backslash begin\; F\backslash colon\; \&\; \backslash mathbb\; F\_q\; \backslash to\; \backslash mathbb\; F\_q\backslash \backslash \; \&\; x\backslash mapsto\; x^p\; \backslash end$
is linear
Linearity is the property of a mathematical relationship (''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out se ...

over $\backslash mathbb\; F\_q,$ and is a field automorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, called the Frobenius automorphism. If $q=p^k,$ the field $\backslash mathbb\; F\_q$ has automorphisms, which are the first powers (under composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

) of . In other words, the Galois group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of $\backslash mathbb\; F\_q$ is cyclic
Cycle or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in social scienc ...

of order , generated by the Frobenius automorphism.
The Diffie–Hellman key exchange
Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include:
* Diffie–Hellman–Merkle key exchange
* Diffie–Hellman key agreement
* Diffie–Hellman key establishment
* Diffie–Hellman key negotiation
* Exponential key exc ...

is an application of exponentiation in finite fields that is widely used for secure communication
Secure communication is when two entities are communicating and do not want a third party to listen in. For this to be the case, the entities need to communicate in a way that is unsusceptible to eavesdropping
Eavesdropping is the act of secret ...

s. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, is computationally expensive. More precisely, if is a primitive element in $\backslash mathbb\; F\_q,$ then $g^e$ can be efficiently computed with exponentiation by squaring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

for any , even if is large, while there is no known algorithm allowing retrieving from $g^e$ if is sufficiently large.
Powers of sets

TheCartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of two sets and is the set of the ordered pair
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s $(x,y)$ such that $x\backslash in\; S$ and $y\backslash in\; T.$ This operation is not properly commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

nor up to Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

canonical
Canonical may refer to:
Science and technology
* Canonical form
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geo ...

isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, that allow identifying, for example, $(x,(y,z)),$ $((x,y),z),$ and $(x,y,z).$
This allows defining the th power $S^n$ of a set as the set of all -tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s $(x\_1,\; \backslash ldots,\; x\_n)$ of elements of .
When is endowed with some structure, it is frequent that $S^n$ is naturally endowed with a similar structure. In this case, the term "direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example $\backslash R^n$ (where $\backslash R$ denotes the real numbers) denotes the Cartesian product of copies of $\backslash R,$ as well as their direct product as vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s, rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

, etc.
Sets as exponents

A -tuple $(x\_1,\; \backslash ldots,\; x\_n)$ of elements of can be considered as afunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

from $\backslash .$ This generalizes to the following notation.
Given two sets and , the set of all functions from to is denoted $S^T$ This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

):
:$(S^T)^U\backslash cong\; S^,$
:$S^\backslash cong\; S^T\backslash times\; S^U,$
where $\backslash times$ denotes the Cartesian product, and $\backslash sqcup$ the disjoint union
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
One can use sets as exponents for other operations on sets, typically for direct sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

s of abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s, vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, or modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...

. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, $\backslash R^\backslash N$ denotes the vector space of the infinite sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

s of real numbers, and $\backslash R^$ the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis
Basis may refer to:
Finance and accounting
*Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items.
Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...

consisting of the sequences with exactly one nonzero element that equals , while the Hamel bases of the former cannot be explicitly described (because there existence involves Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...

).
In this context, can represents the set $\backslash .$ So, $2^S$ denotes the power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of , that is the set of the functions from to $\backslash ,$ which can be identified with the set of the subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of , by mapping each function to the inverse image
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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

In thecategory of sets In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, the morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s between sets and are the functions from to . It results that the set of the functions from to that is denoted $Y^X$ in the preceding section can also be denoted $\backslash hom(X,Y).$ The isomorphism $(S^T)^U\backslash cong\; S^$ can be rewritten
:$\backslash hom(U,S^T)\backslash cong\; \backslash hom(T\backslash times\; U,S).$
This means the functor "exponentiation to the power " is a right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of ...

to the functor "direct product with ".
This generalizes to the definition of exponentiation in a category in which finite direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s exist: in such a category, the functor $X\backslash to\; X^T$ is, if it exists, a right adjoint to the functor $Y\backslash to\; T\backslash times\; Y.$ A category is called a ''Cartesian closed category'', if direct products exist, and the functor $Y\backslash to\; X\backslash times\; Y$ has a right adjoint for every .
Repeated exponentiation

Just 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 ortetration
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. Iterating tetration leads to another operation, and so on, a concept named hyperoperation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. This sequence of operations is expressed by the Ackermann function
In computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes t ...

and Knuth's up-arrow notation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at , the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and () respectively.
Limits of powers

Zero to the power of zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

gives a number of examples of limits that are of the indeterminate formIn calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...

0extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and where the infinities are treated as actual numbers. It is useful in describing the algebra on infiniti ...

, endowed with the product topology
Product may refer to:
Business
* Product (business)
In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...

), which will contain the points at which the function has a limit.
In fact, has a limit at all accumulation point
In mathematics, a limit point (or cluster point or accumulation point) of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood ...

s of , except for , , and . Accordingly, this allows one to define the powers by continuity whenever , , except for 0Efficient computation with integer exponents

Computing ''b''exponentiation by squaring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, where $\backslash sharp\; n$ denotes the number of in the binary representation of . For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiationIn mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. This corresponds to the sequencA003313on the Online Encyclopedia of ...

. Finding the ''minimal'' sequence of multiplications (the minimal-length addition chain for the exponent) for is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' T ...

), but many reasonably efficient heuristic algorithms are available. However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.
Iterated functions

Function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

is a binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

that is defined on functions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

such that the codomain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of the function written on the right is included in the domain
Domain may refer to:
Mathematics
*Domain of a function
In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...

of the function written on the left. It is denoted $g\backslash circ\; f,$ and defined as
:$(g\backslash circ\; f)(x)=g(f(x))$
for every in the domain of .
If the domain of a function equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the th power of the function under composition, commonly called the ''th iterate'' of the function. Thus $f^n$ denotes generally the th iterate of ; for example, $f^3(x)$ means $f(f(f(x))).$
When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, the two kinds of exponentiation are generaly distinguished by placing the exponent of the functional iteration ''before'' the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication ''after'' the parentheses. Thus $f^2(x)=\; f(f(x)),$ and $f(x)^2=\; f(x)\backslash cdot\; f(x).$ When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example $f^=f\backslash circ\; f\; \backslash circ\; f,$ and $f^3=f\backslash cdot\; f\backslash cdot\; f.$ For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. So, $\backslash sin^2\; x$ and $\backslash sin^2(x)$ mean both $\backslash sin(x)\backslash cdot\backslash sin(x)$ and not $\backslash sin(\backslash sin(x)),$ which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.
In this context, the exponent $-1$ denotes always the inverse function
In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...

, if it exists. So $\backslash sin^x=\backslash sin^(x)\; =\; \backslash arcsin\; x.$ For the multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

fractions are generally used as in $1/\backslash sin(x)=\backslash frac\; 1.$
In programming languages

Programming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s generally express exponentiation either as an infix operator or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the caret
The caret () is a V-shaped grapheme, usually inverted and sometimes extended, used in proofreading and typography to indicate that additional material needs to be inserted at this point in the text.
There is a similar mark, , that has a variet ...

(`^`

). The original version of ASCII included an uparrow symbol (`↑`

), intended for exponentiation, but this was replaced by the caret in 1967, so the caret became usual in programming languages.
The notations include:
* `x ^ y`

: AWK
AWK (''awk'') is a domain-specific language
A domain-specific language (DSL) is a computer languageComputer language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (compute ...

, BASIC
BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming language
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the ar ...

, J, MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a and environment developed by . MATLAB allows manipulations, plotting of and data, implementation of s, creation of s, and interfacing with programs written in other languages.
Althoug ...

, Wolfram Language
The Wolfram Language is a general multi-paradigm programming language developed by Wolfram Research. It emphasizes symbolic computation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quant ...

(Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, manipulating Matrix (mathematics), matrices, plotting Fun ...

), R, Microsoft Excel
Microsoft Excel is a spreadsheet
A spreadsheet is a computer application for organization, analysis, and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program ope ...

, Analytica, TeX
TeX (, see below), stylized within the system as TeX, is a typesetting system which was designed and mostly written by Donald Knuth and released in 1978. TeX is a popular means of typesetting complex mathematical formulae; it has been noted ...

(and its derivatives), TI-BASIC
TI-BASIC is the official name of a BASIC
BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming language
In computer science
Computer science deals with the theoretical founda ...

, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua and most computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software
Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data.
It is a type of applica ...

s.
* `x ** y`

. The Fortran
Fortran (; formerly FORTRAN) is a general-purpose, compiled language, compiled imperative programming, imperative programming language that is especially suited to numerical analysis, numeric computation and computational science, scientific com ...

character set did not include lowercase characters or punctuation symbols other than `+-*/()&=.,'`

and so used `**`

for exponentiation. Many other languages followed suit: Ada
Ada may refer to:
Places
Africa
* Ada Foah
Ada Foah is a town on the southeast coast of Ghana, where the Volta River meets the Atlantic Ocean. The town is located along the Volta River, off of the Accra-Aflao motorway. Known for Palm tree, pal ...

, Z shell
The Z shell (Zsh) is a Unix shell
A Unix shell is a command-line interpreter or shell
Shell may refer to:
Architecture and design
* Shell (structure)A shell is a type of structural element which is characterized by its geometry, being a thr ...

, KornShell
KornShell (ksh) is a Unix shell
A Unix shell is a command-line interpreter or shell
Shell may refer to:
Architecture and design
* Shell (structure)A shell is a type of structural element which is characterized by its geometry, being a t ...

, Bash, COBOL
COBOL (; an acronym
An acronym is a word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (lingui ...

, CoffeeScript
CoffeeScript is a programming language that compiles to JavaScript. It adds syntactic sugar inspired by Ruby, Python and Haskell in an effort to enhance JavaScript's brevity and readability. Specific additional features include list comprehens ...

, Fortran
Fortran (; formerly FORTRAN) is a general-purpose, compiled language, compiled imperative programming, imperative programming language that is especially suited to numerical analysis, numeric computation and computational science, scientific com ...

, FoxPro
FoxPro was a text-based
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer ...

, Gnuplot
gnuplot is a command-line
A command-line interface (CLI) processes command
COMMAND.COM is the default command-line interpreter
A command-line interface (CLI) processes commands to a computer program in the form of lines of text. ...

, Groovy
''Groovy'' (or, less commonly, ''groovie'' or ''groovey'') is a slang
Slang is vocabulary (words, phrases, and usage (language), linguistic usages) of an informal register, common in spoken conversation but avoided in formal writing. It also ...

, JavaScript
JavaScript (), often abbreviated JS, is a programming language
A programming language is a formal language
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ma ...

, OCaml
OCaml ( , formerly Objective Caml) is a general-purpose, multi-paradigm programming language
Programming paradigms are a way to classify programming languages based on their features. Languages can be classified into multiple paradigms.
S ...

, F#, Perl
Perl is a family of two high-level
High-level and low-level, as technical terms, are used to classify, describe and point to specific Objective (goal), goals of a systematic operation; and are applied in a wide range of contexts, such as, for ...

, PHP
PHP is a general-purpose scripting language
A scripting language or script language is a programming language
A programming language is a formal language comprising a Instruction set architecture, set of instructions that produce various k ...

, PL/I
PL/I (Programming Language One, pronounced and sometimes written PL/1) is a procedural, imperative
Imperative may refer to:
*Imperative mood, a grammatical mood (or mode) expressing commands, direct requests, and prohibitions
*Imperative prog ...

, Python
Python may refer to:
* Pythonidae
The Pythonidae, commonly known as pythons, are a family of nonvenomous snakes found in Africa, Asia, and Australia. Among its members are some of the largest snakes in the world. Ten genera and 42 species
...

, Rexx
Rexx (Restructured Extended Executor) is a programming language that can be interpreted or compiled
In computing, a compiler is a computer program that Translator (computing), translates computer code written in one programming language (t ...

, Ruby
A ruby is a pink-ish red to blood-red colored gemstone
A gemstone (also called a fine gem, jewel, precious stone, or semi-precious stone) is a piece of mineral
In geology and mineralogy, a mineral or mineral species is, broadly spea ...

, SAS, Seed7
Seed7 is an extensible
Extensibility is a software engineering and systems design
Systems design is the process of defining the architecture
File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Galli ...

, Tcl
Tcl (pronounced "tickle" or as an initialism
An acronym is a word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, prac ...

, ABAP
ABAP (Advanced Business Application Programming, originally ''Allgemeiner Berichts-Aufbereitungs-Prozessor'', German for "general report creation processor") is a high-level programming language
In computer science
Computer science deal ...

, Mercury
Mercury usually refers to:
* Mercury (planet)
Mercury is the smallest planet in the Solar System and the closest to the Sun. Its orbit around the Sun takes 87.97 Earth days, the shortest of all the Sun's planets. It is named after the Roman g ...

, Haskell (for floating-point exponents), Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalysis, cryptanalyst, philosopher, and mathematical and theoretical biology, theoretical biologist. Turing was high ...

, VHDL
The VHSIC Hardware Description Language (VHDL) is a hardware description language
In computer engineering, a hardware description language (HDL) is a specialized computer language used to describe the structure and behavior of electronic ci ...

.
* `x ↑ y`

: Algol Reference language, Commodore BASIC
Commodore BASIC, also known as PET BASIC or CBM-BASIC, is the of the used in 's line, stretching from the of 1977 to the of 1985.
The core is based on , and as such it shares many characteristics with other 6502 BASICs of the time, such ...

, TRS-80 Level II/III BASIC.
* `x ^^ y`

: Haskell (for fractional base, integer exponents), D.
* `x⋆y`

: APL.
In most programming languages with an infix exponentiation operator, it is right-associative
In programming languages
A programming language is a formal language
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

, that is, `a^b^c`

is interpreted as `a^(b^c)`

.Robert W. Sebesta, ''Concepts of Programming Languages'', 2010, , p. 130, 324 This is because `(a^b)^c`

is equal to `a^(b*c)`

and thus not as useful. In some languages, it is left-associative, notably in Algol
Algol , designated Beta Persei (β Persei, abbreviated Beta Per, β Per), known colloquially as the Demon Star, is a bright multiple star in the constellation
A constellation is an area on the celestial s ...

, Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a and environment developed by . MATLAB allows manipulations, plotting of and data, implementation of s, creation of s, and interfacing with programs written in other languages.
Althoug ...

and the Microsoft Excel
Microsoft Excel is a spreadsheet
A spreadsheet is a computer application for organization, analysis, and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program ope ...

formula language.
Other programming languages use functional notation:
* `(expt x y)`

: Common Lisp
Common Lisp (CL) is a dialect of the Lisp programming language
Lisp (historically LISP) is a family of programming language
A programming language is a formal language
In mathematics
Mathematics (from Ancient Greek, Greek: ) incl ...

.
* `pown x y`

: F# (for integer base, integer exponent).
Still others only provide exponentiation as part of standard libraries
A library is a collection of materials, books or media that are easily accessible for use and not just for display purposes. It is responsible for housing updated information in order to meet the user's needs on a daily basis. A library provi ...

:
* `pow(x, y)`

: C, C++
C++ () is a general-purpose programming language
In computer software, a general-purpose programming language is a programming language dedicated to a general-purpose, designed to be used for writing software in a wide variety of application ...

(in `math`

library).
* `Math.Pow(x, y)`

: C#.
* `math:pow(X, Y)`

: Erlang.
* `Math.pow(x, y)`

: Java
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:Pow(x, y)

: PowerShell
PowerShell or Microsoft PowerShell (formerly Windows PowerShell) is a task automation and configuration management program from Microsoft, consisting of a command-line shell (computing), shell and the associated scripting language. Initially a Win ...

.
See also

*Double exponential function
320px, A double exponential function (red curve) compared to a single exponential function (blue curve).
A double exponential function is a constant raised to the power of an exponential function
In mathematics, an exponential function is a ...

* Exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional
Proportionality, proportion or proportional may refer to:
Mathematics
* Proportionality (mathematics), the property of two variables being in a multiplicative rela ...

* Exponential field
* Exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change ...

* List of exponential topics{{short description, Wikipedia list article
This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics.
* Accelerating change
* Mental calculation, Approximating natural exponents (log base e)
* Artin–Hasse exponenti ...

* Modular exponentiation
Modular exponentiation is exponentiation
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and sp ...

* Scientific notation
Scientific notation is a way of expressing numbers
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or coul ...

* Unicode subscripts and superscripts
Unicode
Unicode is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's writing systems. The stan ...

* ''x''Zero to the power of zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

Notes

References

{{Authority control Exponentials Unary operations