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OR:

Exponentiation is a
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, 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 mathematical operations of arithmetic, with the other ones being addi ...
of the base: that is, is the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Pr ...
of multiplying bases: $b^n = \underbrace_.$ The exponent is usually shown as a
superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
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". Starting from the basic fact stated above that, for any positive integer $n$, $b^n$ is $n$ occurrences of $b$ all multiplied by each other, several other properties of exponentiation directly follow. In particular: In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that $b^0$ must be equal to 1, as follows. For any $n$, $b^0 \cdot b^n = b^ = b^n$. Dividing both sides by $b^n$ gives $b^0 = b^n / b^n = 1$. The fact that $b^1 = b$ can similarly be derived from the same rule. For example, $\left(b^1\right)^3 = b^1 \cdot b^1 \cdot b^1 = b^ = b^3$. Taking the cube root of both sides gives $b^1 = b$. The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what $b^$ should mean. In order to respect the "exponents add" rule, it must be the case that $b^ \cdot b^1 = b^ = b^0 = 1$. Dividing both sides by $b^$ gives $b^ = 1 / b^1$, which can be more simply written as $b^ = 1 / b$, using the result from above that $b^1 = b$. By a similar argument, $b^ = 1 / b^n$. The properties of fractional exponents also follow from the same rule. For example, suppose we consider $\sqrt$ and ask if there is some suitable exponent, which we may call $r$, such that $b^r = \sqrt$. From the definition of the square root, we have that $\sqrt \cdot \sqrt = b$. Therefore, the exponent $r$ must be such that $b^r \cdot b^r = b$. Using the fact that multiplying makes exponents add gives $b^ = b$. The $b$ on the right-hand side can also be written as $b^1$, giving $b^ = b^1$. Equating the exponents on both sides, we have $r+r = 1$. Therefore, $r = \frac$, so $\sqrt = b^$. The definition of exponentiation can be extended to allow any real or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. Exponentiation is used extensively in many fields, including
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
,
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary in ...
,
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, ...
,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, and computer science, with applications such as compound interest,
population growth Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...
,
chemical reaction kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
, wave behavior, and
public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic a ...
.

# History of the notation

The term ''power'' ( la, potentia, potestas, dignitas) is a mistranslation of the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
δύναμις (''dúnamis'', here: "amplification") used by the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
mathematician
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
for the square of a line, following
Hippocrates of Chios Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
. In ''
The Sand Reckoner ''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the unive ...
'',
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientist ...
discovered and proved the law of exponents, , necessary to manipulate powers of . In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms مَال (''māl'', "possessions", "property") for a
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
—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 ('' kaʿbah'', "cube") for a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
, which later
Islamic Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God (or ''Allah'') as it was revealed to Muhammad, the ma ...
mathematicians represented in
mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathema ...
as the letters '' mīm'' (m) and ''
kāf Kaph (also spelled kaf) is the eleventh letter of the Semitic abjads, including Phoenician kāp , Hebrew kāf , Aramaic kāp , Syriac kāp̄ , and Arabic kāf (in abjadi order). The Phoenician letter gave rise to the Greek kappa (Κ), Lat ...
'' (k), respectively, by the 15th century, as seen in the work of
Abū al-Hasan ibn Alī al-Qalasādī Abū'l-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qurashī al-Qalaṣādī ( ar, أبو الحسن علي بن محمد بن علي القرشي البسطي; 1412–1486) was a Muslim Arab mathematician from Al-Andalus specializing in Is ...
. In the late 16th century, Jost Bürgi 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. He was born in Erfurt. In 1507 he started to study at the University of Vienna, where he subsequen ...
and
Michael Stifel Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician. He was an Augustinian who became an early supporter of Martin Luther. He was later appointed professor of mathematics at Jena Univer ...
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 () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and last ...
used the terms square, cube, zenzizenzic (
fourth power In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So: :''n''4 = ''n'' × ''n'' × ''n'' × ''n'' Fourth powers are also formed by multiplying a number by its cube. Furthe ...
), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (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: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathe ...
in his text titled '' La Géométrie''; 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, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the ...
) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
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, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
introduced variable exponents, and, implicitly, non-integer exponents by writing:

# Terminology

The expression is called "the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
of ''b''" or "''b'' squared", because the area of a square with side-length is . Similarly, the expression is called "the
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
of ''b''" or "''b'' cubed", because the volume of a cube with side-length is . When it is a
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, 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 elementary arithmetic operations.

## Positive exponents

The definition of the exponentiation as an iterated multiplication can be formalized by using induction, and this definition can be used as soon one has an
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
multiplication: The base case is :$b^1 = b$ and the recurrence is :$b^ = b^n \cdot b.$ The associativity of multiplication implies that for any positive integers and , :$b^ = b^m \cdot b^n,$ and :$\left(b^m\right)^n=b^.$

## 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 :$b^=b^m\cdot b^n$ to zero exponents. It may be used in every
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
with a multiplication that has an identity. Intuitionally, $b^0$ may be interpreted as the
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in questio ...
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 : :$b^ = \frac$. Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity ($\infty$). This definition of exponentiation with negative exponents is the only one that allows extending the identity $b^=b^m\cdot b^n$ to negative exponents (consider the case $m=-n$). The same definition applies to invertible elements in a multiplicative
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
, that is, an
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
, with an associative multiplication and a
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
denoted (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element is standardly denoted $x^.$

## Identities and properties

The following identities, often called , hold for all integer exponents, provided that the base is non-zero: :$\begin b^ &= b^m \cdot b^n \\ \left\left(b^m\right\right)^n &= b^ \\ \left(b \cdot c\right)^n &= b^n \cdot c^n \end$ Unlike addition and multiplication, exponentiation is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. For example, . Also unlike addition and multiplication, exponentiation is not
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
. For example, , whereas . Without parentheses, the conventional
order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For examp ...
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 :$\left\left(b^p\right\right)^q = b^ .$

## Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by the
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
:$\left(a+b\right)^n=\sum_^n \binoma^ib^=\sum_^n \fraca^ib^.$ 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, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
that is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. Otherwise, if and are, say, square matrices of the same size, this formula cannot be used. It follows that in
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
, many
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
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 with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. Th ...
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 of functions from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of elements to a set of elements (see cardinal exponentiation). Such functions can be represented as - tuples 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 (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic num ...
) 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 that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, ...
to denote large or small numbers. For instance, (the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
in vacuum, in
metres per second The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre in ...
) 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 pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
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 and Pueblo, Colorado. It also streams online. History KLST and KPIK-FM The 94.3 signal signed on the air on August 22, 1962, as KLST, owned by Little London Br ...
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 ( : halves) is the irreducible fraction resulting from dividing one by two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or "halving"; conversely, ...
'' and '' quarter''. Powers of appear in
set theory Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
, since a set with members has a
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
, the set of all of its
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s, which has members. Integer powers of are important in computer science. The positive integer powers give the number of possible values for an -
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented ...
integer
binary number A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
; for example, a
byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable un ...
may take different values. The binary number system expresses any number as a sum of powers of , and denotes it as a sequence of and , separated by a binary point, 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 alternating
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s. For a similar discussion of powers of the complex number , see .

## Large exponents

The
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
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, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
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 form In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
, are described in below.

## Power functions  Real functions of the form $f\left(x\right) = cx^n$, where $c \ne 0$, are sometimes called power functions. When $n$ is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
and $n \ge 1$, two primary families exist: for $n$ even, and for $n$ odd. In general for $c > 0$, when $n$ is even $f\left(x\right) = cx^n$ will tend towards positive
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions a ...
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 grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
are called even functions. When $n$ is odd, $f\left(x\right)$'s
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
behavior reverses from positive $x$ to negative $x$. For $c > 0$, $f\left(x\right) = cx^n$ will also tend towards positive
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions a ...
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.

# Rational exponents

If is a nonnegative
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, and is a positive integer, $x^$ or 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 $\frac pq$ is a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
, with and integers, then $x^$ is defined as :$x^\frac pq= \left\left(x^p\right\right)^\frac 1q=\left(x^\frac 1q\right)^p.$ The equality on the right may be derived by setting $y=x^\frac 1q,$ and writing $\left(x^\frac 1q\right)^p=y^p=\left\left(\left(y^p\right)^q\right\right)^\frac 1q=\left\left(\left(y^q\right)^p\right\right)^\frac 1q=\left(x^p\right)^\frac 1q.$ If is a positive rational number, $0^r=0,$ by definition. All these definitions are required for extending the identity $\left(x^r\right)^s = x^$ 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 $x^\frac 1n,$ the identity $\left(x^a\right)^b=x^$ cannot be satisfied. For example, :$\left\left(\left(-1\right)^2\right\right)^\frac 12 = 1^\frac 12= 1\neq \left(-1\right)^ =\left(-1\right)^1=-1.$ 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 the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of the base and the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
(, 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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
, but there is no choice of the principal value for which the identity :$\left\left(b^r\right\right)^s = b^$ is true; see . Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a
multivalued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
.

## Limits of rational exponents Since any
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
can be expressed as the
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
of rational numbers, exponentiation of a positive real number with an arbitrary real exponent can be defined by continuity with the rule :$b^x = \lim_ b^r \quad \left(b \in \mathbb^+,\, x \in \mathbb\right),$ 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 In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain $b^\pi:$ : So, the upper bounds and the lower bounds of the intervals form two
sequences In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
that have the same limit, denoted $b^\pi.$ This defines $b^x$ for every positive and real as a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...

## The exponential function

The ''exponential function'' is often defined as $x\mapsto e^x,$ where $e\approx 2.718$ is
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an express ...
. For avoiding
circular reasoning Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet) A flyer (or flier) is a form of paper Paper is a thin sheet material pro ...
, this definition cannot be used here. So, a definition of the exponential function, denoted $\exp\left(x\right),$ 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 :$\exp\left(x\right)=e^x.$ There are many equivalent ways to define the exponential function, one of them being :$\exp\left(x\right) = \lim_ \left\left(1 + \frac\right\right)^n.$ One has $\exp\left(0\right)=1,$ and the ''exponential identity'' $\exp\left(x+y\right)=\exp\left(x\right)\exp\left(y\right)$ holds as well, since :$\exp\left(x\right)\exp\left(y\right) = \lim_ \left\left(1 + \frac\right\right)^n\left\left(1 + \frac\right\right)^n = \lim_ \left\left(1 + \frac + \frac\right\right)^n,$ and the second-order term $\frac$ does not affect the limit, yielding $\exp\left(x\right)\exp\left(y\right) = \exp\left(x+y\right)$. Euler's number can be defined as $e=\exp\left(1\right)$. It follows from the preceding equations that $\exp\left(x\right)=e^x$ when is an integer (this results from the repeated-multiplication definition of the exponentiation). If is real, $\exp\left(x\right)=e^x$ 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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
value of , and therefore it can be used to extend the definition of $\exp\left(z\right)$, and thus $e^z,$ from the real numbers to any complex argument . This extended exponential function still satisfies 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 and
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
function. Specifically, the fact that the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, ...
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 ...
of the exponential function means that one has : $b = \exp\left(\ln b\right)=e^$ for every . For preserving the identity $\left(e^x\right)^y=e^,$ one must have :$b^x=\left\left(e^ \right\right)^x = e^$ 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 and
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
exponent is defined by means of the exponential function with complex argument (see the end of , above) as :$b^z = e^,$ where $\ln b$ denotes the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, ...
of . This satisfies the identity :$b^ = b^z b^t,$ In general, $\left(b^z\right)^t$ 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, :$\left\left(b^z\right\right)^t \ne b^,$ unless is real or is an integer.
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
, :$e^ = \cos y + i \sin y,$ allows expressing the
polar form In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of $b^z$ in terms of the real and imaginary parts of , namely :$b^= b^x\left(\cos\left(y\ln b\right)+i\sin\left(y\ln b\right)\right),$ where the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
of the
trigonometric Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
factor is one. This results from :$b^=b^x b^=b^x e^ =b^x\left(\cos\left(y\ln b\right)+i\sin\left(y\ln b\right)\right).$

# 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 use
complex logarithm In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to be ...
s, and is therefore easier to understand.

## th roots of a complex number

Every nonzero complex number may be written in
polar form In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
as :$z=\rho e^=\rho\left(\cos \theta +i \sin \theta\right),$ where $\rho$ is the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
of , and $\theta$ is its
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
. The argument is defined
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R' ...
an integer multiple of ; this means that, if $\theta$ is the argument of a complex number, then $\theta +2k\pi$ 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 : : If $2\pi$ is added to $\theta$, the complex number is not changed, but this adds $2i\pi/n$ 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 $-\pi<\theta\le \pi,$ 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, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
in the whole complex plane, except for negative real values of the radicand. 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 In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
shows that the principal th root is the unique complex differentiable 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 $2\pi,$ the complex number comes back to its initial position, and its th roots are permuted circularly (they are multiplied by $e^$). This shows that it is not possible to define a th root function that is 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 in
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a compl ...
or algebraic solutions of algebraic equations ( Lagrange resolvent). The th roots of unity are the first powers of $\omega =e^\frac$, that is $1=\omega^0=\omega^n, \omega=\omega^1, \omega^2, \omega^.$ The th roots of unity that have this generating property are called ''primitive th roots of unity''; they have the form $\omega^k=e^\frac,$ with
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
at the vertices of a regular -gon with one vertex on the real number 1. As the number $e^\frac$ is the primitive th root of unity with the smallest positive
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
, 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, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
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 a
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
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, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
. In all cases, the
complex logarithm In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to be ...
is used to define complex exponentiation as :$z^w=e^,$ where $\log z$ is the variant of the complex logarithm that is used, which is, a function or a
multivalued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
such that :$e^=z$ for every in its domain of definition.

### Principal value

The
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
of the
complex logarithm In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to be ...
is the unique function, commonly denoted $\log,$ such that, for every nonzero complex number , :$e^=z,$ and the
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of satisfies :$-\pi <\mathrm \le \pi.$ The principal value of the complex logarithm is not defined for $z=0,$ it is discontinuous at negative real values of , and it is holomorphic (that is, complex differentiable) elsewhere. If is real and positive, the principal value of the complex logarithm is the natural logarithm: $\log z=\ln z.$ The principal value of $z^w$ is defined as $z^w=e^,$ where $\log z$ is the principal value of the logarithm. The function $\left(z,w\right)\to z^w$ 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 $\log z$ and $z^w$ at the negative real values of . In this case, it is useful to consider these functions as
multivalued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
s. If $\log z$ denotes one of the values of the multivalued logarithm (typically its principal value), the other values are $2ik\pi +\log z,$ 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
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
, that is, there is an integer such that is an integer. This results from the
periodicity Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values tha ...
of the exponential function, more specifically, that $e^a=e^b$ if and only if $a-b$ is an integer multiple of $2\pi i.$ If $w=\frac mn$ is a rational number with and
coprime integers In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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 $z\ne 0,$ 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 disc ...
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, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of ''. If $z=a+ib$ is the canonical form of ( and being real), then its polar form is $z=\rho e^= \rho (\cos\theta + i \sin\theta),$ where $\rho=\sqrt$ and $\theta=\operatorname\left(a,b\right)$ (see
atan2 In computing and mathematics, the function atan2 is the 2- argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive
for the definition of this function). *''
Logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of ''. The
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
of this logarithm is $\log z=\ln \rho+i\theta,$ where $\ln$ denotes the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, ...
. The other values of the logarithm are obtained by adding $2ik\pi$ for any integer . *''Canonical form of $w\log z.$'' If $w=c+di$ with and real, the values of $w\log z$ are $w\log z = (c\ln \rho - d\theta-2dk\pi) +i (d\ln \rho + c\theta+2ck\pi),$ the principal value corresponding to $k=0.$ *''Final result.'' Using the identities $e^=e^xe^y$ and $e^ =x^y,$ one gets $z^w=\rho^c e^ \left(\cos (d\ln \rho + c\theta+2ck\pi) +i\sin(d\ln \rho + c\theta+2ck\pi)\right),$ with $k=0$ for the principal value.

### =Examples

= * $i^i$
The polar form of is $i=e^,$ and the values of $\log i$ are thus $\log i=i\left(\frac \pi 2 +2k\pi\right).$ It follows that $i^i=e^=e^ e^.$So, all values of $i^i$ are real, the principal one being $e^ \approx 0.2079.$ *$\left(-2\right)^$
Similarly, the polar form of is $-2 = 2e^.$ So, the above described method gives the values $\begin (-2)^ &= 2^3 e^ (\cos(4\ln 2 + 3(\pi +2k\pi)) +i\sin(4\ln 2 + 3(\pi+2k\pi)))\\ &=-2^3 e^(\cos(4\ln 2) +i\sin(4\ln 2)). \end$In this case, all the values have the same argument $4\ln 2,$ 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 an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
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 real
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the ...
, and is a rational number, then is an algebraic number. This results from the theory of
algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ex ...
s. This remains true if is any algebraic number, in which case, all values of (as a
multivalued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
) are algebraic. If is
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. T ...
(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 $b\not\in \,$ 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 any associative operation denoted as a multiplication.More generally, power associativity 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, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
. An
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
. 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 In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
. In this case, the inverse of is denoted $x^,$ and $x^n$ is defined as $\left\left(x^\right\right)^.$ Exponentiation with integer exponents obeys the following laws, for and in the algebraic structure, and and integers: :$\begin x^0&=1\\ x^&=x^m x^n\\ \left(x^m\right)^n&=x^\\ \left(xy\right)^n&=x^n y^n \quad \text xy=yx, \text \end$ These definitions are widely used in many areas of mathematics, notably for
groups A group is a number of persons 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 ide ...
, rings, fields, square matrices (which form a ring). They apply also to functions from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to itself, which form a monoid under function composition. This includes, as specific instances,
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
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 gr ...
s of any
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
. 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, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
whose valued can be multiplied, $f^n$ denotes the exponentiation with respect of multiplication, and $f^$ may denote exponentiation with respect of function composition. That is, :$\left(f^n\right)\left(x\right)=\left(f\left(x\right)\right)^n=f\left(x\right) \,f\left(x\right) \cdots f\left(x\right),$ and :$\left(f^\right)\left(x\right)=f\left(f\left(\cdots f\left(f\left(x\right)\right)\cdots\right)\right).$ Commonly, $\left(f^n\right)\left(x\right)$ is denoted $f\left(x\right)^n,$ while $\left(f^\right)\left(x\right)$ is denoted $f^n\left(x\right).$

## In a group

A
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred ...
is a set with as associative operation denoted as multiplication, that has an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
, and such that every element has an inverse. So, if is a group, $x^n$ is defined for every $x\in G$ 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 ''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, ''H'' is a subgro ...
. A group (or subgroup) that consists of all powers of a specific element is the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative ...
generated by . If all the powers of are distinct, the group is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word 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 structur ...
$\Z$ of the integers. Otherwise, the cyclic group is
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
(it has a finite number of elements), and its number of elements is the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
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 abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen a ...
. 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 In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fix ...
), and in the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or el ...
. 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 $\left(g^h\right)^k=g^$ and $\left(gh\right)^k=g^kh^k.$

## In a ring

In a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, it may occur that some nonzero elements satisfy $x^n=0$ for some integer . Such an element is said to be
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
. In a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, the nilpotent elements form an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
, called the nilradical of the ring. If the nilradical is reduced to the zero ideal (that is, if $x\neq 0$ implies $x^n\neq 0$ for every positive integer ), the commutative ring is said reduced. Reduced rings important in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometric ...
, since the coordinate ring of an affine algebraic set is always a reduced ring. More generally, given an ideal in a commutative ring , the set of the elements of that have a power in is an ideal, called the
radical Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
of . The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
).

## 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 $A^ = \left\left(A^\right\right)^n$. 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 a
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happ ...
, 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 transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
. Apart from matrices, more general
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s can also be exponentiated. An example is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
operator of calculus, $d/dx$, which is a linear operator acting on functions $f\left(x\right)$ to give a new function $\left(d/dx\right)f\left(x\right) = f\text{'}\left(x\right)$. The ''n''-th power of the differentiation operator is the ''n''-th derivative: :$\left\left(\frac\right\right)^nf\left(x\right) = \fracf\left(x\right) = f^\left(x\right).$ 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 together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
. 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 and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
,
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
,
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and se ...
, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the
fractional calculus 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 differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...
.

## Finite fields

A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
and every nonzero element has a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
. 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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s and their subfields, the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s and the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
. 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 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 because the only wa ...
or a
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
; 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, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, which allows, in general, working as if there were only one field with elements, denoted $\mathbb F_q.$ One has :$x^q=x$ for every $x\in \mathbb F_q.$ A primitive element in $\mathbb F_q$ is an element such the set of the first powers of (that is, $\$) equals the set of the nonzero elements of $\mathbb F_q.$ There are $\varphi \left(p-1\right)$ primitive elements in $\mathbb F_q,$ where $\varphi$ is
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
. In $\mathbb F_q,$ 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) and x,y are nonzero real numbers. Beginning students commonly make this error in computi ...
identity :$\left(x+y\right)^p = x^p+y^p$ is true for the exponent . As $x^p=x$ in $\mathbb F_q,$ It follows that the map :$\begin F\colon & \mathbb F_q \to \mathbb F_q\\ & x\mapsto x^p \end$ is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
over $\mathbb F_q,$ and is a
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
, called the Frobenius automorphism. If $q=p^k,$ the field $\mathbb F_q$ has automorphisms, which are the first powers (under
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
) of . In other words, the
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the ...
of $\mathbb F_q$ is cyclic 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 or interception. Secure communicatio ...
s. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, if is a primitive element in $\mathbb F_q,$ then $g^e$ can be efficiently computed with exponentiation by squaring for any , even if is large, while there is no known algorithm allowing retrieving from $g^e$ if is sufficiently large.

# Powers of sets

The
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of two sets and is the set of the ordered pairs $\left(x,y\right)$ such that $x\in S$ and $y\in T.$ This operation is not properly
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
nor
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, but has these properties
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R' ...
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s, that allow identifying, for example, $\left(x,\left(y,z\right)\right),$ $\left(\left(x,y\right),z\right),$ and $\left(x,y,z\right).$ This allows defining the th power $S^n$ of a set as the set of all - tuples $\left(x_1, \ldots, x_n\right)$ of elements of . When is endowed with some structure, it is frequent that $S^n$ is naturally endowed with a similar structure. In this case, the term " direct product" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example $\R^n$ (where $\R$ denotes the real numbers) denotes the Cartesian product of copies of $\R,$ as well as their direct product as
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
,
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s, rings, etc.

## Sets as exponents

A -tuple $\left(x_1, \ldots, x_n\right)$ of elements of can be considered as a function from $\.$ 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 and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f tha ...
): :$\left(S^T\right)^U\cong S^,$ :$S^\cong S^T\times S^U,$ where $\times$ denotes the Cartesian product, and $\sqcup$ the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
. One can use sets as exponents for other operations on sets, typically for
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
s of
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 sy ...
. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, $\R^\N$ denotes the vector space of the
infinite sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
s of real numbers, and $\R^$ 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 basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
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). In this context, can represents the set $\.$ So, $2^S$ denotes the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of , that is the set of the functions from to $\,$ which can be identified with the set of the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of , by mapping each function to the inverse image of . This fits in with the exponentiation of cardinal numbers, in the sense that , where is the cardinality of .

## In category theory

In the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
, the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
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 $\hom\left(X,Y\right).$ The isomorphism $\left(S^T\right)^U\cong S^$ can be rewritten :$\hom\left(U,S^T\right)\cong \hom\left(T\times U,S\right).$ 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 exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
to the functor "direct product with ". This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor $X\to X^T$ is, if it exists, a right adjoint to the functor $Y\to T\times Y.$ A category is called a ''Cartesian closed category'', if direct products exist, and the functor $Y\to X\times Y$ 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 or
tetration In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as re ...
. Iterating tetration leads to another operation, and so on, a concept named
hyperoperation In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with the ...
. This sequence of operations is expressed by the
Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are tot ...
and Knuth's up-arrow notation. 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 Zero to the power of zero, denoted by , is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines  . In mathematical analysis, the expression is so ...
gives a number of examples of limits that are of the
indeterminate form In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
00. 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 $f\left(x,y\right) = x^y$ defined on $D = \$. Then can be viewed as a subset of (that is, the set of all pairs with , belonging to the
extended 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 -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
, endowed with the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
), 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, accumulation point, or cluster point of a 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 of x with respect to the topology on X also contai ...
s of , except for , , and . Accordingly, this allows one to define the powers by continuity whenever , , except for 00, (+∞)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.

# Efficient computation with integer exponents

Computing ''b''''n'' using iterated multiplication requires multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, apply Horner's rule to the exponent 100 written in binary: :$100 = 2^2 +2^5 + 2^6 = 2^2\left(1+2^3\left(1+2\right)\right)$. 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 $\sharp n +\lfloor \log_ n\rfloor -1,$ by using exponentiation by squaring, where $\sharp n$ denotes the number of in the
binary representation A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
of . For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiation. Finding the ''minimal'' sequence of multiplications (the minimal-length addition chain for the exponent) for is a difficult problem, for which no efficient algorithms are currently known (see
Subset sum problem 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''.'' ...
), 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 is a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary ...
that is defined on functions such that the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either t ...
of the function written on the right is included in the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
of the function written on the left. It is denoted $g\circ f,$ and defined as :$\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$ 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, $f^3\left(x\right)$ means $f\left(f\left(f\left(x\right)\right)\right).$ 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, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the func ...
, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration ''before'' the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication ''after'' the parentheses. Thus $f^2\left(x\right)= f\left(f\left(x\right)\right),$ and $f\left(x\right)^2= f\left(x\right)\cdot f\left(x\right).$ When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example $f^=f\circ f \circ f,$ and $f^3=f\cdot f\cdot f.$ For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
. So, $\sin^2 x$ and $\sin^2\left(x\right)$ both mean $\sin\left(x\right)\cdot\sin\left(x\right)$ and not $\sin\left(\sin\left(x\right)\right),$ 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 (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
, if it exists. So $\sin^x=\sin^\left(x\right) = \arcsin x.$ For the
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
fractions are generally used as in $1/\sin\left(x\right)=\frac 1.$

# In programming languages

Programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming l ...
s generally express exponentiation either as an infix
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another s ...
or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the
caret Caret is the name used familiarly for the character , provided on most QWERTY keyboards by typing . The symbol has a variety of uses in programming and mathematics. The name "caret" arose from its visual similarity to the original proofrea ...
(^). 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,
BASIC BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming languages designed for ease of use. The original version was created by John G. Kemeny and Thomas E. Kurtz at Dartmouth Colleg ...
, J,
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
,
Wolfram Language The Wolfram Language ( ) is a general multi-paradigm programming language developed by Wolfram Research. It emphasizes symbolic computation, functional programming, and rule-based programming and can employ arbitrary structures and data. It is ...
(
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiz ...
), R, Microsoft Excel, Analytica,
TeX Tex may refer to: People and fictional characters * Tex (nickname), a list of people and fictional characters with the nickname * Joe Tex (1933–1982), stage name of American soul singer Joseph Arrington Jr. Entertainment * ''Tex'', the Italian ...
(and its derivatives),
TI-BASIC TI-BASIC is the official name of a BASIC-like language built into Texas Instruments (TI)'s graphing calculators. TI-BASIC is a language family of three different and incompatible versions, released on different products: * TI-BASIC 83 (on Z80 ...
, 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 with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. Th ...
s. * x ** y. The Fortran character set did not include lowercase characters or punctuation symbols other than +-*/()&=.,' and so used ** for exponentiation (the initial version used a xx b instead.). Many other languages followed suit: Ada,
Z shell The Z shell (Zsh) is a Unix shell that can be used as an interactive login shell and as a command interpreter for shell scripting. Zsh is an extended Bourne shell with many improvements, including some features of Bash, ksh, and tcsh. H ...
, KornShell, Bash, COBOL,
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 compreh ...
, Fortran,
FoxPro FoxPro was a text-based procedurally oriented programming language and database management system (DBMS), and it was also an object-oriented programming language, originally published by Fox Software and later by Microsoft, for MS-DOS, Window ...
,
Gnuplot gnuplot is a command-line and GUI program that can generate two- and three-dimensional plots of functions, data, and data fits. The program runs on all major computers and operating systems (Linux, Unix, Microsoft Windows, macOS, FreeDO ...
,
Groovy ''Groovy'' (or, less commonly, ''groovie'' or ''groovey'') is a slang colloquialism popular during the 1950s, '60s and '70s. It is roughly synonymous with words such as "excellent", "fashionable", or "amazing", depending on context. History The ...
,
JavaScript JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of websites use JavaScript on the client side for webpage behavior, often ...
,
OCaml OCaml ( , formerly Objective Caml) is a general-purpose, multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Di ...
, F#,
Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offi ...
,
PHP PHP is a general-purpose scripting language geared toward web development. It was originally created by Danish-Canadian programmer Rasmus Lerdorf in 1993 and released in 1995. The PHP reference implementation is now produced by The PHP Group. ...
,
PL/I PL/I (Programming Language One, pronounced and sometimes written PL/1) is a procedural, imperative computer programming language developed and published by IBM. It is designed for scientific, engineering, business and system programming. I ...
, Python,
Rexx Rexx (Restructured Extended Executor) is a programming language that can be interpreted or compiled. It was developed at IBM by Mike Cowlishaw. It is a structured, high-level programming language designed for ease of learning and reading. ...
, Ruby,
SAS SAS or Sas may refer to: Arts, entertainment, and media * ''SAS'' (novel series), a French book series by Gérard de Villiers * '' Shimmer and Shine'', an American animated children's television series * Southern All Stars, a Japanese rock ...
,
Seed7 Seed7 is an extensible general-purpose programming language designed by Thomas Mertes. It is syntactically similar to Pascal and Ada. Along with many other features, it provides an extension mechanism. Daniel Zingaro"Modern Extensible Languag ...
,
Tcl TCL or Tcl or TCLs may refer to: Business * TCL Technology, a Chinese consumer electronics and appliance company ** TCL Electronics, a subsidiary of TCL Technology * Texas Collegiate League, a collegiate baseball league * Trade Centre Limite ...
,
ABAP ABAP (Advanced Business Application Programming, originally ''Allgemeiner Berichts-Aufbereitungs-Prozessor'', German for "general report preparation processor") is a high-level programming language created by the German software company SAP SE. ...
,
Mercury Mercury commonly refers to: * Mercury (planet), the nearest planet to the Sun * Mercury (element), a metallic chemical element with the symbol Hg * Mercury (mythology), a Roman god Mercury or The Mercury may also refer to: Companies * Mercu ...
Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical com ...
, VHDL. * x ↑ y: Algol Reference language, Commodore BASIC, TRS-80 Level II/III BASIC. * x ^^ y: Haskell (for fractional base, integer exponents), D. * x⋆y: APL. In most programming languages with an infix exponentiation operator, it is
right-associative In programming language theory, the associativity of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators (for example, ...
, 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 (; short for "Algorithmic Language") is a family of imperative computer programming languages originally developed in 1958. ALGOL heavily influenced many other languages and was the standard method for algorithm description used by the ...
,
Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
and the Microsoft Excel formula language. Other programming languages use functional notation: * (expt x y):
Common Lisp Common Lisp (CL) is a dialect of the Lisp programming language, published in ANSI standard document ''ANSI INCITS 226-1994 (S20018)'' (formerly ''X3.226-1994 (R1999)''). The Common Lisp HyperSpec, a hyperlinked HTML version, has been derived f ...
. * 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 accessible for use and not just for display purposes. A library provides physical (hard copies) or digital access (soft copies) materials, and may be a physical location or a vi ...
: * pow(x, y): C,
C++ C, or c, is the third Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Le ...
(in math library). * Math.Pow(x, y): C#. * math:pow(X, Y): Erlang. * Math.pow(x, y):
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
. * ath Ath (; nl, Aat, ; pcd, Ât; wa, Ate) is a city and municipality of Wallonia located in the province of Hainaut, Belgium. The municipality consists of the following districts: Arbre, Ath, Bouvignies, Ghislenghien, Gibecq, Houtaing ...:Pow(x, y):
PowerShell PowerShell is a task automation and configuration management program from Microsoft, consisting of a command-line shell and the associated scripting language. Initially a Windows component only, known as Windows PowerShell, it was made open-s ...
.

* Double exponential function *
Exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ( lambda) is a positive ra ...
* 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) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
*
List of exponential topics {{Short description, none This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. * Accelerating change * Approximating natural exponents (log base e) * Artin–Hasse exponential * Bacterial growth * B ...
* Modular exponentiation *
Scientific notation Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, ...
*
Unicode subscripts and superscripts Unicode has subscripted and superscripted versions of a number of characters including a full set of Arabic numerals. These characters allow any polynomial, chemical and certain other equations to be represented in plain text without using an ...
* ''x''''y'' = ''y''''x'' *
Zero to the power of zero Zero to the power of zero, denoted by , is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines  . In mathematical analysis, the expression is so ...

# References

{{Authority control Exponentials Unary operations