View source for Exponentiation

In [[Mathematics|mathematics]], '''exponentiation''', denoted {{math|''b''<sup>''n''</sup>}}, is an [[Operation (mathematics)|operation]] involving two numbers: the ''base'', {{mvar|b}}, and the ''exponent'' or ''power'', {{mvar|n}}.<ref name=":1">{{cite web |last=Nykamp |first=Duane |title=Basic rules for exponentiation |website=Math Insight |url=https://mathinsight.org/exponentiation_basic_rules |access-date=August 27, 2020}}</ref> When {{mvar|n}} is a positive [[Integer|integer]], exponentiation corresponds to repeated [[Multiplication|multiplication]] of the base: that is, {{math|''b''<sup>''n''</sup>}} is the [[Product (mathematics)|product]] of multiplying {{mvar|n}} bases:<ref name=":1"/>
<math display="block">b^n = \underbrace{b \times b \times \dots \times b \times b}_{n \text{ times}}.</math>In particular, <math>b^1=b</math>.

The exponent is usually shown as a superscript to the right of the base as {{math|''b''<sup>''n''</sup>}} or in computer code as <code>b^n</code>. This [[Binary operation|binary operation]] is often read as "{{mvar|b}} to the power {{mvar|n}}"; it may also be referred to as "{{mvar|b}} raised to the {{mvar|n}}th power", "the {{mvar|n}}th power of {{mvar|b}}",<ref>{{MathWorld |title=Power |id=Power |access-date=2020-08-27}}</ref> or, most briefly, "{{mvar|b}} to the {{mvar|n}}".

The above definition of <math>b^n</math> immediately implies several properties, in particular the multiplication rule:<ref group="nb">There are three common notations for [[Multiplication|multiplication]]: <math>x\times y</math> is most commonly used for explicit numbers and at a very elementary level; <math>xy</math> is most common when [[Variable (mathematics)|variable]]s are used; <math>x\cdot y</math> is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.</ref>

<math display="block">\begin{align}
b^n \times b^m & = \underbrace{b \times \dots \times b}_{n \text{ times}} \times \underbrace{b \times \dots \times b}_{m \text{ times}}  \\[1ex]
&  = \underbrace{b \times \dots \times b}_{n+m \text{ times}} \ =\ b^{n+m} . 
\end{align}</math>

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add. Extending this rule to the power zero gives <math>b^0 \times b^n = b^{0+n} = b^n</math>, and, where {{mvar|b}} is non-zero, dividing both sides by <math>b^n</math> gives <math>b^0 = b^n / b^n = 1</math>. That is the multiplication rule implies the definition <math display="block">
b^0=1.
</math>A similar argument implies the definition for negative integer powers: <math display="block">b^{-n} = 1/b^n.</math>That is, extending the multiplication rule gives <math>b^{-n} \times b^n = b^{-n+n} = b^0 = 1 </math>. Dividing both sides by <math>b^n</math> gives <math>b^{-n} = 1 / b^n</math>. This also implies the definition for fractional powers: <math display="block">b^{n/m} = \sqrt[m]{b^n}.</math>For example, <math> b^{1/2} \times b^{1/2} = b^{1/2 \,+\, 1/2} = b^1 = b  </math>, meaning <math> (b^{1/2})^2 = b </math>, which is the definition of square root: <math>b^{1/2} = \sqrt{b} </math>.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define <math>b^x</math> for any positive real base <math>b</math> and any real number exponent <math>x</math>. More involved definitions allow complex base and exponent, as well as certain types of [[Matrix (mathematics)|matrices]] as base or exponent.

Exponentiation is used extensively in many fields, including [[Finance:Economics|economics]], [[Biology:Biology|biology]], [[HandWiki:Chemistry|chemistry]], [[HandWiki:Physics|physics]], and [[Computer science|computer science]], with applications such as [[Compound interest|compound interest]], population growth, chemical reaction kinetics, [[Wave|wave]] behavior, and [[Public-key cryptography|public-key cryptography]].

==Etymology==
The term ''exponent'' originates from the [[Social:Latin|Latin]] ''exponentem'', the present participle of ''exponere'', meaning "to put forth".<ref>{{cite web |title=Exponent &#124; Etymology of exponent by etymonline |url=https://www.etymonline.com/word/exponent}}</ref> The term ''power'' ({{langx|la|potentia, potestas, dignitas}}) is a mistranslation<ref name="Rotman">{{cite book |last=Rotman |first=Joseph J. |date=2015 |title=Advanced Modern Algebra, Part 1 |location=Providence, RI |publisher=American Mathematical Society |at=p. 130, fn. 4 |isbn=978-1-4704-1554-9 |edition=3rd |series=[[Graduate Studies in Mathematics]] |volume=165 |url=https://www.ams.org/books/gsm/165/04 |url-access=subscription}}</ref><ref>{{cite book |last=Szabó |first=Árpád |date=1978 |title=The Beginnings of Greek Mathematics |location=Dordrecht |publisher=D. Reidel |page=[https://archive.org/details/TheBeginningsOfGreekMathematics/page/n37 37] |isbn=90-277-0819-3 |series=Synthese Historical Library |volume=17 |translator=A.M. Ungar |url=https://archive.org/details/TheBeginningsOfGreekMathematics}}</ref> of the [[Social:Ancient Greek|ancient Greek]] δύναμις (''dúnamis'', here: "amplification"<ref name="Rotman"/>) used by the Greek mathematician [[Biography:Euclid|Euclid]] for the square of a line,<ref name="MacTutor"/> following [[Biography:Hippocrates of Chios|Hippocrates of Chios]].<ref>{{cite book |last=Ball |first=W. W. Rouse |author-link=W. W. Rouse Ball |date=1915 |title=A Short Account of the History of Mathematics |location=London |publisher=Macmillan |page=[https://archive.org/details/shortaccountofhi00ballrich/page/38 38] |edition=6th |url=https://archive.org/details/shortaccountofhi00ballrich}}</ref>

The word ''exponent'' was coined in 1544 by Michael Stifel.<ref>{{cite web|url=https://jeff560.tripod.com/e.html|title=Earliest Known Uses of Some of the Words of Mathematics (E)|date=June 23, 2017|url-status=dead|archive-date=15 September 2024|archive-url=https://web.archive.org/web/20240915223031/https://jeff560.tripod.com/e.html}}</ref><ref>{{cite book |last=Stifel |first=Michael |date=1544 |title=Arithmetica integra |location=Nuremberg |publisher=Johannes Petreius |page=235v |url=https://archive.org/details/bub_gb_fndPsRv08R0C/page/n491}}</ref> In the 16th century, [[Biography:Robert Recorde|Robert Recorde]] used the terms "square", "cube", "zenzizenzic" ([[Fourth power|fourth power]]), "sursolid" ([[Fifth power (algebra)|fifth]]), "zenzicube" ([[Sixth power|sixth]]), "second sursolid" ([[Seventh power|seventh]]), and "[[Zenzizenzizenzic|zenzizenzizenzic]]" ([[Eighth power|eighth]]).<ref name="worldwidewords">{{cite web |title=Zenzizenzizenzic |publisher=World Wide Words |last=Quinion |first=Michael |author-link=Michael Quinion |url=https://www.worldwidewords.org/weirdwords/ww-zen1.htm |access-date=2020-04-16}}</ref> "Biquadrate" has been used to refer to the fourth power as well.

==History==
In ''[[Astronomy:The Sand Reckoner|The Sand Reckoner]]'', [[Biography:Archimedes|Archimedes]] proved the law of exponents, {{math|1=10<sup>''a''</sup> · 10<sup>''b''</sup> = 10<sup>''a''+''b''</sup>}}, necessary to manipulate powers of {{math|10}}.<ref>
Archimedes. (2009). THE SAND-RECKONER. In T. Heath (Ed.), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters (Cambridge Library Collection - Mathematics, pp. 229-232). Cambridge: Cambridge University Press. {{doi|10.1017/CBO9780511695124.017}}.</ref> He then used powers of {{math|10}} to estimate the number of grains of sand that can be contained in the universe.

In the 9th century, the Persian mathematician Al-Khwarizmi used the terms مَال (''māl'', "possessions", "property") for a [[Square (algebra)|square]]—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"<ref name="worldwidewords"/>—and كَعْبَة (''[[Religion:Kaaba|Kaʿbah]]'', "cube") for a [[Cube (algebra)|cube]], which later Islamic mathematicians represented in [[Mathematical notation|mathematical notation]] as the letters ''mīm'' (m) and ''kāf'' (k), respectively, by the 15th century, as seen in the work of [[Biography:Abu'l-Hasan ibn Ali al-Qalasadi|Abu'l-Hasan ibn Ali al-Qalasadi]].<ref>{{MacTutor|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref>
[[Biography:Nicolas Chuquet|Nicolas Chuquet]] used a form of exponential notation in the 15th century, for example {{math|12<sup>2</sup>}} to represent {{math|12''x''<sup>2</sup>}}.<ref>{{cite book |last=Cajori |first=Florian |title=A History of Mathematical Notations |date=1928 |publisher=The Open Court Company |volume=1 |page=102 |url=https://archive.org/details/historyofmathema031756mbp}}</ref> This was later used by Henricus Grammateus and [[Biography:Michael Stifel|Michael Stifel]] in the 16th century. In the late 16th century, [[Biography:Jost Bürgi|Jost Bürgi]] would use Roman numerals for exponents in a way similar to that of Chuquet, for example {{Overset|iii|4}} for {{math|4''x''<sup>3</sup>}}.<ref name="cajori">{{cite book |last=Cajori |first=Florian |date=1928 |title=A History of Mathematical Notations |location=London |publisher=[[Company:Open Court Publishing Company|Open Court Publishing Company]] |page=[https://archive.org/details/historyofmathema031756mbp/page/n363 344] |volume=1 |url=https://archive.org/details/historyofmathema031756mbp}}</ref>

In 1636, James Hume used in essence modern notation, when in ''L'algèbre de Viète'' he wrote {{math|''A''<sup>iii</sup>}} for {{math|''A''<sup>3</sup>}}.<ref>{{cite book |last=Cajori |first=Florian |title=A History of Mathematical Notations |date=1928 |publisher=The Open Court Company |volume=1 |pages=204 |url=https://archive.org/details/historyofmathema031756mbp}}</ref> Early in the 17th century, the first form of our modern exponential notation was introduced by [[Biography:René Descartes|René Descartes]] in his text titled ''[[La Géométrie]]''; there, the notation is introduced in Book I.<ref>{{cite book |last=Descartes |first=René |date=1637 |title=Discourse de la méthode [...] |location=Leiden |publisher=Jan Maire |page=299 |chapter=''[[La Géométrie]]'' |url=http://gallica.bnf.fr/ark:/12148/btv1b86069594/f383.image |quote=Et ''aa'', ou {{math|''a''<sup>2</sup>}}, pour multiplier {{math|''a''}} par soy mesme; Et {{math|''a''<sup>3</sup>}}, pour le multiplier encore une fois par {{math|''a''}}, & ainsi a l'infini}} (And {{math|''aa''}}, or {{math|''a''<sup>2</sup>}}, in order to multiply {{math|''a''}} by itself; and {{math|''a''<sup>3</sup>}}, in order to multiply it once more by {{math|''a''}}, and thus to infinity).</ref>

{{Blockquote|text=I designate ... {{math|''aa''}}, or {{math|''a''<sup>2</sup>}} in multiplying {{math|''a''}} by itself; and {{math|''a''<sup>3</sup>}} in multiplying it once more again by {{math|''a''}}, and thus to infinity.|author=René Descartes|title=[[La Géométrie]]}}

Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write [[Polynomial|polynomial]]s, for example, as {{math|''ax'' + ''bxx'' + ''cx''<sup>3</sup> + ''d''}}.

Samuel Jeake introduced the term ''indices'' in 1696.<ref name="MacTutor">{{MacTutor|class=Miscellaneous|id=Mathematical_notation|title=Etymology of some common mathematical terms}}</ref> The term ''involution'' was used synonymously with the term ''indices'', but had declined in usage<ref>The most recent usage in this sense cited by the OED is from 1806 ({{cite OED |involution}}).</ref> and should not be confused with [[Involution (mathematics)|its more common meaning]].

In 1748, [[Biography:Leonhard Euler|Leonhard Euler]] introduced variable exponents, and, implicitly, non-integer exponents by writing:{{blockquote|Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not [[Algebraic function|algebraic function]]s, since in those the exponents must be constant.<ref name="Euler_1748"/>}}

=== 20th century ===
As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For example [[Biography:Konrad Zuse|Konrad Zuse]] introduced [[Floating-point arithmetic|floating-point arithmetic]] in his 1938 computer Z1. One register contained representation of leading digits, and a second contained representation of the exponent of 10. Earlier Leonardo Torres Quevedo contributed ''Essays on Automation'' (1914) which had suggested the floating-point representation of numbers. The more flexible decimal floating-point representation was introduced in 1946 with a Bell Laboratories computer. Eventually educators and engineers adopted [[Scientific notation|scientific notation]] of numbers, consistent with common reference to [[Order of magnitude|order of magnitude]] in a [[Ratio scale|ratio scale]].<ref>Janet Shiver & Terri Wiilard "[https://www.visionlearning.com/en/library/Math-in-Science/62/Scientific-Notation/250 Scientific notation: working with orders of magnitude] from Visionlearning</ref>

For instance, in 1961 the [[School Mathematics Study Group]] developed the notation in connection with units used in the [[Metric system|metric system]].<ref>School Mathematics Study Group (1961) ''Mathematics for Junior High School'', volume 2, part 1, Yale University Press</ref><ref>Cecelia Callanan (1967) "Scientific Notation", ''The Mathematics Teacher'' 60: 252–6 [https://www.jstor.org/stable/27957540 JSTOR]</ref>

Exponents also came to be used to describe units of measurement and quantity dimensions. For instance, since [[Physics:Force|force]] is mass times acceleration, it is measured in kg m/sec<sup>2</sup>. Using M for mass, L for length, and T for time, the expression M L T<sup>–2</sup> is used in [[Dimensional analysis|dimensional analysis]] to describe force.<ref>[[Biography:Edwin Bidwell Wilson|Edwin Bidwell Wilson]] (1920) [https://archive.org/details/aeronauticsclass00wilsrich/page/182/mode/2up Theory of Dimensions], chapter 11 in ''Aeronautics: A Class Text'', via Internet Archive</ref><ref>{{cite book |last1= Bridgman|first1= Percy Williams|title= Dimensional Analysis|publisher= Yale University Press|location= New Haven|date= 1922|oclc= 840631|url= https://archive.org/details/dimensionalanaly00bridrich }}</ref>

==Terminology==
The expression {{math|1=''b''<sup>2</sup> = ''b'' · ''b''}} is called "the [[Square (algebra)|square]] of {{math|''b''}}" or "{{math|''b''}} squared", because the area of a square with side-length {{math|''b''}} is {{math|''b''<sup>2</sup>}}. (It is true that it could also be called "{{math|''b''}} to the second power", but "the square of {{math|''b''}}" and "{{math|''b''}} squared" are more traditional)

Similarly, the expression {{math|1=''b''<sup>3</sup> = ''b'' · ''b'' · ''b''}} is called "the [[Cube (algebra)|cube]] of {{math|''b''}}" or "{{math|''b''}} cubed", because the volume of a cube with side-length {{math|''b''}} is {{math|''b''<sup>3</sup>}}.

When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, {{math|1=3<sup>5</sup> = 3 · 3 · 3 · 3 · 3 = 243}}. The base {{math|3}} appears {{math|5}} times in the multiplication, because the exponent is {{math|5}}. Here, {{math|243}} 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 {{math|3<sup>5</sup>}} can be simply read "3 to the 5th", or "3 to the 5".

== Integer exponents <span class="anchor" id="Integer"></span> ==

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 [[Formal proof|formalized]] by using [[Mathematical induction|induction]],<ref>{{cite book |title=Abstract Algebra: an inquiry based approach |last1=Hodge |first1=Jonathan K. |last2=Schlicker |first2=Steven |last3=Sundstorm |first3=Ted |page=94 |date=2014 |publisher=CRC Press |isbn=978-1-4665-6706-1 |url=https://books.google.com/books?id=qToTAgAAQBAJ&pg=PA94}}</ref> and this definition can be used as soon as one has an [[Associativity|associative]] multiplication:

The base case is
: <math>b^1 = b</math>
and the [[Recurrence relation|recurrence]] is
: <math>b^{n+1} = b^n \cdot b.</math>

The associativity of multiplication implies that for any positive integers {{mvar|m}} and {{mvar|n}},
: <math>b^{m+n} = b^m \cdot b^n,</math>
and
: <math>(b^m)^n=b^{mn}.</math>

===Zero exponent===
As mentioned earlier, a (nonzero) number raised to the {{math|0}} power is {{math|1}}:<ref>{{cite book |title=Technical Shop Mathematics |last1=Achatz |first1=Thomas |page=101 |date=2005 |edition=3rd |publisher=Industrial Press |isbn=978-0-8311-3086-2 |url=https://books.google.com/books?id=YOdtemSmzQQC&pg=PA101}}</ref><ref name=":1"/>
: <math>b^0=1.</math>

This value is also obtained by the [[Empty product|empty product]] convention, which may be used in every [[Algebraic structure|algebraic structure]] with a multiplication that has an identity. This way the formula
: <math>b^{m+n}=b^m\cdot b^n</math>
also holds for <math>n=0</math>.

The case of {{math|0<sup>0</sup>}} is controversial. In contexts where only integer powers are considered, the value {{math|1}} is generally assigned to {{math|0<sup>0</sup>}} but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. {{Crossreference|For more details, see [[Zero to the power of zero]].}}

===Negative exponents===
Exponentiation with negative exponents is defined by the following identity, which holds for any integer {{mvar|n}} and nonzero {{mvar|b}}:<ref name=":1"/>
: <math>b^{-n} = \frac{1}{b^n}</math>
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (<math>\infty</math>).<ref>{{cite book
 | editor1 = Kostas Gavroglu |editor2=Jean Christianidis |editor3=Efthymios Nicolaidis
 | contribution = The infinite in Leibniz’s mathematics – The historiographical method of comprehension in context
 | doi = 10.1007/978-94-017-3596-4_20
 | isbn = 9789401735964
 | publisher = Springer Netherlands
 | series = Boston Studies in the Philosophy of Science
 | title = Trends in the Historiography of Science
 | year = 1994
 | volume = 151 |page = 276
 |quote=A positive power of zero is infinitely small, a negative power of zero is infinite.}}</ref>

This definition of exponentiation with negative exponents is the only one that allows extending the identity <math>b^{m+n}=b^m\cdot b^n</math> to negative exponents (consider the case <math>m=-n</math>).

The same definition applies to [[Invertible element|invertible element]]s in a multiplicative [[Monoid|monoid]], that is, an [[Algebraic structure|algebraic structure]], with an associative multiplication and a multiplicative identity denoted {{math|1}} (for example, the [[Square matrix|square matrices]] of a given dimension). In particular, in such a structure, the inverse of an [[Invertible element|invertible element]] {{mvar|x}} is standardly denoted <math>x^{-1}.</math>

===Identities and properties===
{{Redirect|Laws of Indices|the horse|Laws of Indices (horse)}}
The following [[Identity (mathematics)|identities]], often called '''{{vanchor|exponent rules}}''', hold for all integer exponents, provided that the base is non-zero:<ref name=":1"/>
: <math>\begin{align}
           b^m \cdot b^n &= b^{m + n} \\
  \left(b^m\right)^n &= b^{m \cdot n} \\
       b^n \cdot c^n &= (b \cdot c)^n
\end{align}</math>

Unlike addition and multiplication, exponentiation is not commutative: for example, <math>2^3 = 8</math>, but reversing the operands gives the different value <math>3^2=9</math>. Also unlike addition and multiplication, exponentiation is not associative: for example, {{math|1=(2<sup>3</sup>)<sup>2</sup> = 8<sup>2</sup> = 64}}, whereas {{math|1=2<sup>(3<sup>2</sup>)</sup> = 2<sup>9</sup> = 512}}. Without parentheses, the conventional [[Order of operations|order of operations]] for serial exponentiation in superscript notation is top-down (or ''right''-associative), not bottom-up<ref name="Bronstein_1987"/><ref name="NIST_2010"/><ref name="Zeidler_2013"/> (or ''left''-associative). That is,
: <math>b^{p^q} = b^{\left(p^q\right)},</math>
which, in general, is different from
: <math>\left(b^p\right)^q = b^{p q} .</math>

===Powers of a sum===
The powers of a sum can normally be computed from the powers of the summands by the binomial formula
: <math>(a+b)^n=\sum_{i=0}^n \binom{n}{i}a^ib^{n-i}=\sum_{i=0}^n \frac{n!}{i!(n-i)!}a^ib^{n-i}.</math>

However, this formula is true only if the summands commute (i.e. that {{math|1=''ab'' = ''ba''}}), which is implied if they belong to a [[Algebraic structure|structure]] that is [[Commutative property|commutative]]. Otherwise, if {{mvar|a}} and {{mvar|b}} are, say, [[Square matrix|square matrices]] of the same size, this formula cannot be used. It follows that in [[Computer algebra|computer algebra]], many [[Algorithm|algorithm]]s involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose [[Software:Computer algebra system|computer algebra system]]s use a different notation (sometimes {{math|^^}} instead of {{math|^}}) for exponentiation with non-commuting bases, which is then called '''non-commutative exponentiation'''.

===Combinatorial interpretation===

For nonnegative integers {{mvar|n}} and {{mvar|m}}, the value of {{math|''n''<sup>''m''</sup>}} is the number of [[Function (mathematics)|functions]] from a [[Set (mathematics)|set]] of {{mvar|m}} elements to a set of {{mvar|n}} elements (see cardinal exponentiation). Such functions can be represented as {{mvar|m}}-[[Tuple|tuple]]s from an {{mvar|n}}-element set (or as {{mvar|m}}-letter words from an {{mvar|n}}-letter alphabet). Some examples for particular values of {{mvar|m}} and {{mvar|n}} are given in the following table:

{| class="wikitable"
!{{math|''n''<sup>''m''</sup>}}
!The {{math|''n''<sup>''m''</sup>}} possible {{mvar|m}}-tuples of elements from the set {{math|{{mset|1, ..., ''n''}}}}
|-
|0{{sup|5}} = 0
|{{CNone|none}}
|-
|1{{sup|4}} = 1
|(1, 1, 1, 1)
|-
|2{{sup|3}} = 8
|(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)
|-
|3{{sup|2}} = 9
|(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
|-
|4{{sup|1}} = 4
|(1), (2), (3), (4)
|-
|5{{sup|0}} = 1
|()
|}

===Particular bases===

====Powers of ten <span class="anchor" id="Base 10"></span>====
{{See also|Scientific notation}}
{{Main|Power of 10}}
In the base ten ([[Decimal|decimal]]) number system, integer powers of {{math|10}} are written as the digit {{math|1}} followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, {{math|1={{val|e=3}} = {{val|1000}}}} and {{math|1={{val|e=-4}} = {{val|0.0001}}}}.

Exponentiation with base {{math|[[10 (number)|10]]}} is used in [[Scientific notation|scientific notation]] to denote large or small numbers. For instance, {{val|299792458|u=m/s}} (the [[Physics:Speed of light|speed of light]] in vacuum, in metres per second) can be written as {{val|2.99792458|e=8|u=m/s}} and then [[Approximation|approximated]] as {{val|2.998|e=8|u=m/s}}.

[[SI prefix]]es based on powers of {{math|10}} are also used to describe small or large quantities. For example, the prefix [[Kilo-|kilo]] means {{math|1={{val|e=3}} = {{val|1000}}}}, so a kilometre is {{val|1000|u=metres}}.

===={{anchor|Base 2}}Powers of two====
{{Main|Power of two}}
The first negative powers of {{math|2}} have special names: <math>2^{-1}</math>is a ''[[One half|half]]''; <math>2^{-2}</math> is a ''[[4 (number)|quarter]].''

Powers of {{math|2}} appear in [[Set theory|set theory]], since a set with {{math|''n''}} members has a [[Power set|power set]], the set of all of its [[Subset|subset]]s, which has {{math|2<sup>''n''</sup>}} members.

Integer powers of {{math|2}} are important in [[Computer science|computer science]]. The positive integer powers {{math|2<sup>''n''</sup>}} give the number of possible values for an {{math|''n''}}-[[Bit|bit]] integer [[Binary number|binary number]]; for example, a [[Byte|byte]] may take {{math|1=2<sup>8</sup> = 256}} different values. The binary number system expresses any number as a sum of powers of {{math|2}}, and denotes it as a sequence of {{math|0}} and {{math|1}}, separated by a binary point, where {{math|1}} indicates a power of {{math|2}} that appears in the sum; the exponent is determined by the place of this {{math|1}}: the nonnegative exponents are the rank of the {{math|1}} on the left of the point (starting from {{math|0}}), and the negative exponents are determined by the rank on the right of the point.

====Powers of one====
Every power of one equals: {{math|1=1<sup>''n''</sup> = 1}}.

====Powers of zero====
For a positive exponent {{math|''n'' > 0}}, the {{mvar|n}}th power of zero is zero: {{math|1=0<sup>''n''</sup> = 0}}. For a negative exponent, <math>0^{-n}=1/0^n=1/0</math> is undefined.

In some contexts (e.g., [[Combinatorics|combinatorics]]), the expression [[zero to the power of zero|{{math|0<sup>0</sup>}}]] is defined to be equal to <math>1</math>; in others (e.g., [[Mathematical analysis|analysis]]), it is often undefined.

====Powers of negative one====
Since a negative number times another negative is positive, we have:<blockquote><math>(-1)^n = \left\{\begin{array}{rl} 
1 & \text{for even } n, \\
-1 & \text{for odd } n. \\
\end{array}\right.</math></blockquote>Because of this, powers of {{math|−1}} are useful for expressing alternating [[Sequence|sequence]]s. For a similar discussion of powers of the complex number {{math|''i''}}, see {{slink||nth roots of a complex number}}.

===Large exponents===
The [[Limit of a sequence|limit of a sequence]] of powers of a number greater than one diverges; in other words, the sequence grows without bound:
: {{math|''b''<sup>''n''</sup> → ∞}} as {{math|''n'' → ∞}} when {{math|''b'' > 1}}

This can be read as "''b'' to the power of ''n'' tends to [[Extended real number line|+∞]] as ''n'' tends to infinity when ''b'' is greater than one".

Powers of a number with [[Absolute value|absolute value]] less than one tend to zero:
: {{math|''b''<sup>''n''</sup> → 0}} as {{math|''n'' → ∞}} when {{math|{{abs|''b''}} < 1}}

Any power of one is always one:
: {{math|1=''b''<sup>''n''</sup> = 1}} for all {{math|''n''}} for {{math|1=''b'' = 1}}

Powers of a negative number <math>b\leq -1</math> alternate between positive and negative as {{math|''n''}} alternates between even and odd, and thus do not tend to any limit as {{math|''n''}} grows.

If the exponentiated number varies while tending to {{math|1}} as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
: {{math|(1 + 1/''n'')<sup>''n''</sup> → ''e''}} as {{math|''n'' → ∞}}

See {{slink||Exponential function}} below.

Other limits, in particular those of expressions that take on an [[Indeterminate form|indeterminate form]], are described in {{slink||Limits of powers}} below.

===Power functions===
{{Main|Power law}}

[[File:Potenssi 1 3 5.svg|thumb|left|Power functions for {{math|1=''n'' = 1, 3, 5}}]]
[[File:Potenssi 2 4 6.svg|thumb|Power functions for {{math|1=''n'' = 2, 4, 6}}]]

Real functions of the form <math>f(x) = cx^n</math>, where <math>c \ne 0</math>, are sometimes called power functions.<ref>{{cite book |last1=Hass |first1=Joel R. |last2=Heil |first2=Christopher E. |last3=Weir |first3=Maurice D. |last4=Thomas |first4=George B. |title=Thomas' Calculus |date=2018 |publisher=Pearson |isbn=9780134439020 |pages=7–8 |edition=14}}</ref> When <math>n</math> is an [[Integer|integer]] and <math>n \ge 1</math>, two primary families exist: for <math>n</math> even, and for <math>n</math> odd. In general for <math>c > 0</math>, when <math>n</math> is even <math>f(x) = cx^n</math> will tend towards positive infinity with increasing <math>x</math>, and also towards positive infinity with decreasing <math>x</math>. All graphs from the family of even power functions have the general shape of <math>y=cx^2</math>, flattening more in the middle as <math>n</math> increases.<ref name="Calculus: Early Transcendentals">{{cite book |last1=Anton |first1=Howard |last2=Bivens |first2=Irl |last3=Davis |first3=Stephen |title=Calculus: Early Transcendentals |date=2012 |publisher=John Wiley & Sons |page=[https://archive.org/details/calculusearlytra00anto_656/page/n51 28] |isbn=9780470647691 |edition=9th |url=https://archive.org/details/calculusearlytra00anto_656 |url-access=limited}}</ref> Functions with this kind of [[Symmetry|symmetry]] {{nobr|(<math>f(-x)= f(x)</math>)}} are called even functions.

When <math>n</math> is odd, <math>f(x)</math>'s asymptotic behavior reverses from positive <math>x</math> to negative <math>x</math>. For <math>c > 0</math>, <math>f(x) = cx^n</math> will also tend towards positive infinity with increasing <math>x</math>, but towards negative infinity with decreasing <math>x</math>. All graphs from the family of odd power functions have the general shape of <math>y=cx^3</math>, flattening more in the middle as <math>n</math> increases and losing all flatness there in the straight line for <math>n=1</math>. Functions with this kind of symmetry {{nobr|(<math>f(-x)= -f(x)</math>)}} are called odd functions.

For <math>c < 0</math>, the opposite asymptotic behavior is true in each case.<ref name="Calculus: Early Transcendentals"/>

===Table of powers===
{|class="wikitable" style="text-align:right"
! ''b'' !! ''b''<sup>2</sup> !! ''b''<sup>3</sup> !! ''b''<sup>4</sup> !! ''b''<sup>5</sup> !! ''b''<sup>6</sup> !! ''b''<sup>7</sup> !! ''b''<sup>8</sup> !! ''b''<sup>9</sup> !! ''b''<sup>10</sup>  !! ''b''<sup>''n''</sup> for {{calculator label|''n''|for=n}} = {{calculator|id=n|size=3|type=number|default=11}} ||[[On-Line Encyclopedia of Integer Sequences|OEIS]] number
|-
|'''1''' || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || {{OEIS link|A000012}}
|-
|'''2''' || 4 || 8 || 16 || 32 || 64 || 128 || 256 || 512 || {{val|1,024|fmt=gaps}}|| {{calculator|type=plain|formula=pow(2,n)|default=2048|mapping={"overflow": "Infinity"} }} || {{OEIS link| A000079}}
|-
|'''3''' || 9 || 27 || 81 || 243 || 729 || {{val|2,187|fmt=gaps}} || {{val|6,561|fmt=gaps}} || {{val|19,683}} || {{val|59,049}} || {{calculator|type=plain|formula=pow(3,n)|default=177147|mapping={"overflow": "Infinity"} }} || {{OEIS link|A000244}}
|-
|'''4''' || 16 || 64 || 256 || {{val|1,024|fmt=gaps}} || {{val|4,096|fmt=gaps}} || {{val|16,384}} || {{val|65,536}} || {{val|262,144}} || {{val|1,048,576}} || {{calculator|type=plain|formula=pow(4,n)|default=4194304|mapping={"overflow": "Infinity"} }} || {{OEIS link|A000302}}
|-
|'''5''' || 25 || 125 || 625 || {{val|3,125|fmt=gaps}} || {{val|15,625}} || {{val|78,125}} || {{val|390,625}} || {{val|1,953,125}} || {{val|9,765,625}} || {{calculator|type=plain|formula=pow(5,n)|default=48828125|mapping={"overflow": "Infinity"} }} || {{OEIS link|A000351}}
|-
|'''6''' || 36 || 216 || {{val|1,296|fmt=gaps}} || {{val|7,776|fmt=gaps}} || {{val|46,656}} || {{val|279,936}} || {{val|1,679,616}} || {{val|10,077,696}} || {{val|60,466,176}} || {{calculator|type=plain|formula=pow(6,n)|default=362797056|mapping={"overflow": "Infinity"} }} || {{OEIS link|A000400}}
|-
|'''7''' || 49 || 343 || {{val|2,401|fmt=gaps}} || {{val|16,807}} || {{val|117,649}} || {{val|823,543}} || {{val|5,764,801}} || {{val|40,353,607}} || {{val|282,475,249}} || {{calculator|type=plain|formula=pow(7,n)|default=1977326743|mapping={"overflow": "Infinity"} }} || {{OEIS link|A000420}}
|-
|'''8''' || 64 || 512 || {{val|4,096|fmt=gaps}} || {{val|32,768}} || {{val|262,144}} || {{val|2,097,152}} || {{val|16,777,216}} || {{val|134,217,728}} || {{val|1,073,741,824}} || {{calculator|type=plain|formula=pow(8,n)|default=8589934592|mapping={"overflow": "Infinity"} }} || {{OEIS link|A001018}}
|-
|'''9''' || 81 || 729 || {{val|6,561|fmt=gaps}} || {{val|59,049}} || {{val|531,441}} || {{val|4,782,969}} || {{val|43,046,721}} || {{val|387,420,489}} || {{val|3,486,784,401}} || {{calculator|type=plain|formula=pow(9,n)|default=31381059609|mapping={"overflow": "Infinity"} }} || {{OEIS link|A001019}}
|-
|'''10''' || 100 || {{val|1,000|fmt=gaps}} || {{val|10,000}} || {{val|100,000}} || {{val|1,000,000}} || {{val|10,000,000}} || {{val|100,000,000}} || {{val|1,000,000,000}} || {{val|10,000,000,000}} || {{calculator|type=plain|formula=pow(10,n)|default=100000000000|mapping={"overflow": "Infinity"} }} || {{OEIS link|A011557}}
|-
| {{calculator|id=b|size=6|type=number|default=11}}
| {{calculator|type=plain|formula=pow(b,2)|default=121}}
| {{calculator|type=plain|formula=pow(b,3)|default=1331}}
| {{calculator|type=plain|formula=pow(b,4)|default=14641}}
| {{calculator|type=plain|formula=pow(b,5)|default=161051}}
| {{calculator|type=plain|formula=pow(b,6)|default=1771561}}
| {{calculator|type=plain|formula=pow(b,7)|default=19487171}}
| {{calculator|type=plain|formula=pow(b,8)|default=214358881}}
| {{calculator|type=plain|formula=pow(b,9)|default=2357947691}}
| {{calculator|type=plain|formula=pow(b,10)|default=25937424601}}
| {{calculator|type=plain|formula=pow(b,n)|default=285311670611|mapping={"overflow": "Infinity"} }}
|
|-
| '''[[On-Line Encyclopedia of Integer Sequences|OEIS]] number''' || {{OEIS link|A000290}} || {{OEIS link|A000578}} || {{OEIS link|A000583}} || {{OEIS link|A000584}} || {{OEIS link|A001014}} || {{OEIS link|A001015}} || {{OEIS link|A001016}} || {{OEIS link|A001017}} || {{OEIS link|A008454}} || ||
|}

==Rational exponents==
[[File:Mplwp roots 01.svg|thumb|From top to bottom: {{math|''x''<sup>1/8</sup>}}, {{math|''x''<sup>1/4</sup>}}, {{math|''x''<sup>1/2</sup>}}, {{math|''x''<sup>1</sup>}}, {{math|''x''<sup>2</sup>}}, {{math|''x''<sup>4</sup>}}, {{math|''x''<sup>8</sup>}}.]]

If {{mvar|x}} is a nonnegative [[Real number|real number]], and {{mvar|n}} is a positive integer, <math>x^{1/n}</math> or <math>\sqrt[n]x</math> denotes the unique nonnegative real [[nth root|{{mvar|n}}th root]] of {{mvar|x}}, that is, the unique nonnegative real number {{mvar|y}} such that <math>y^n=x.</math>

If {{mvar|x}} is a positive real number, and <math>\frac pq</math> is a [[Rational number|rational number]], with {{mvar|p}} and {{mvar|q > 0}} integers, then <math display="inline">x^{p/q}</math> is defined as
:<math>x^\frac pq= \left(x^p\right)^\frac 1q=(x^\frac 1q)^p.</math>
The equality on the right may be derived by setting <math>y=x^\frac 1q,</math> and writing <math>(x^\frac 1q)^p=y^p=\left((y^p)^q\right)^\frac 1q=\left((y^q)^p\right)^\frac 1q=(x^p)^\frac 1q.</math>

If {{mvar|r}} is a positive rational number, {{math|1=0<sup>''r''</sup> = 0}}, by definition.

All these definitions are required for extending the identity <math>(x^r)^s = x^{rs}</math> 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 {{mvar|n}}th root, which is negative, if {{mvar|n}} is [[Odd number|odd]], and no real root if {{mvar|n}} is even. In the latter case, whichever complex {{mvar|n}}th root one chooses for <math>x^\frac 1n,</math> the identity <math>(x^a)^b=x^{ab}</math> cannot be satisfied. For example,
:<math>\left((-1)^2\right)^\frac 12 = 1^\frac 12= 1\neq (-1)^{2\cdot\frac 12} =(-1)^1=-1.</math>

See {{slink||Real exponents}} and {{slink||Non-integer exponents with a complex base}} 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 ({{slink||Limits of rational exponents}}, below), or in terms of the [[Logarithm|logarithm]] of the base and the [[Exponential function|exponential function]] ({{slink||Powers via logarithms}}, 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 number|complex]] 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. One may choose one of these values, called the [[Principal value|principal value]], but there is no choice of the principal value for which the identity
: <math>\left(b^r\right)^s = b^{r s}</math>
is true; see {{slink||Failure of power and logarithm identities}}. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a [[Multivalued function|multivalued function]].

===Limits of rational exponents===
[[File:Continuity of the Exponential at 0.svg|thumb|The limit of {{math|''e''{{sup|1/''n''}}}} is {{math|1=''e''{{sup|0}} = 1}} when {{mvar|n}} tends to the infinity.]]
Since any [[Irrational number|irrational number]] can be expressed as the [[Limit of a sequence|limit of a sequence]] of rational numbers, exponentiation of a positive real number {{mvar|b}} with an arbitrary real exponent {{mvar|x}} can be defined by [[Continuous function|continuity]] with the rule<ref name="Denlinger">{{cite book |title=Elements of Real Analysis |last=Denlinger |first=Charles G. |publisher=Jones and Bartlett |date=2011 |pages=278–283 |isbn=978-0-7637-7947-4}}</ref>
:<math> b^x = \lim_{r (\in \mathbb{Q}) \to x} b^r \quad (b \in \mathbb{R}^+,\, x \in \mathbb{R}),</math>
where the limit is taken over rational values of {{mvar|r}} only. This limit exists for every positive {{mvar|b}} and every real {{mvar|x}}.

For example, if {{math|1=''x'' = {{pi}}}}, the non-terminating decimal representation {{math|1=''π'' = 3.14159...}} and the [[Monotone function|monotonicity]] of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain <math>b^\pi:</math>
:<math>\left[b^3, b^4\right], \left[b^{3.1}, b^{3.2}\right], \left[b^{3.14}, b^{3.15}\right], \left[b^{3.141}, b^{3.142}\right], \left[b^{3.1415}, b^{3.1416}\right], \left[b^{3.14159}, b^{3.14160}\right], \ldots</math>
So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted <math>b^\pi.</math>

This defines <math>b^x</math> for every positive {{mvar|b}} and real {{mvar|x}} as a [[Continuous function|continuous function]] of {{mvar|b}} and {{mvar|x}}. See also [[Well-defined expression]].<ref>{{cite book |chapter=Limits of sequences |chapter-url={{Google books |ecTsDAAAQBAJ |page=154 |plainurl=yes}} |title=Analysis I |series=Texts and Readings in Mathematics |year=2016 |last1=Tao |first1=Terence |volume=37 |pages=126–154 |isbn=978-981-10-1789-6 |doi=10.1007/978-981-10-1789-6_6}}</ref>

===Exponential function===
{{Main|Exponential function}}
The ''exponential function'' may be defined as <math>x\mapsto e^x,</math> where <math>e\approx 2.718</math> is Euler's number, but to avoid [[Philosophy:Circular reasoning|circular reasoning]], this definition cannot be used here. Rather, we give an independent definition of the exponential function <math>\exp(x),</math> and of <math>e=\exp(1)</math>, relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition: <math>\exp(x)=e^x.</math>

There are [[Characterizations of the exponential function|many equivalent ways to define the exponential function]], one of them being
: <math>\exp(x) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n.</math>

One has <math>\exp(0)=1,</math> and the ''exponential identity'' (or multiplication rule) <math>\exp(x)\exp(y)=\exp(x+y)</math> holds as well, since
: <math>\exp(x)\exp(y) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n\left(1 + \frac{y}{n}\right)^n = \lim_{n\rightarrow\infty} \left(1 + \frac{x+y}{n} + \frac{xy}{n^2}\right)^n,</math>
and the second-order term <math>\frac{xy}{n^2}</math> does not affect the limit, yielding <math>\exp(x)\exp(y) = \exp(x+y)</math>.

Euler's number can be defined as <math>e=\exp(1)</math>. It follows from the preceding equations that <math>\exp(x)=e^x</math> when {{mvar|x}} is an integer (this results from the repeated-multiplication definition of the exponentiation). If {{mvar|x}} is real, <math>\exp(x)=e^x</math> results from the definitions given in preceding sections, by using the exponential identity if {{mvar|x}} is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every [[Complex number|complex]] value of {{mvar|x}}, and therefore it can be used to extend the definition of <math>\exp(z)</math>, and thus <math>e^z,</math> from the real numbers to any complex argument {{mvar|z}}. 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 {{math|''e''<sup>''x''</sup>}} as the exponential function allows defining {{math|''b''<sup>''x''</sup>}} for every positive real numbers {{mvar|b}}, in terms of exponential and [[Logarithm|logarithm]] function. Specifically, the fact that the [[Natural logarithm|natural logarithm]] {{math|ln(''x'')}} is the [[Inverse function|inverse]] of the exponential function {{math|''e''<sup>''x''</sup>}} means that one has
: <math>b = \exp(\ln b)=e^{\ln b}</math>
for every {{math|''b'' > 0}}. For preserving the identity <math>(e^x)^y=e^{xy},</math> one must have
: <math>b^x=\left(e^{\ln b} \right)^x = e^{x \ln b}</math>

So, <math>e^{x \ln b}</math> can be used as an alternative definition of {{math|''b''<sup>''x''</sup>}} for any positive real {{mvar|b}}. 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 {{mvar|b}} is a positive real number, exponentiation with base {{mvar|b}} and [[Complex number|complex]] exponent {{mvar|z}} is defined by means of the exponential function with complex argument (see the end of {{slink||Exponential function}}, above) as
: <math>b^z = e^{(z\ln b)},</math>
where <math>\ln b</math> denotes the [[Natural logarithm|natural logarithm]] of {{mvar|b}}.

This satisfies the identity
: <math>b^{z+t} = b^z b^t,</math>
In general,
<math DISPLAY=inline>\left(b^z\right)^t</math> is not defined, since {{math|''b''<sup>''z''</sup>}} is not a real number. If a meaning is given to the exponentiation of a complex number (see {{slink||Non-integer exponents with a complex base}}, below), one has, in general,
: <math>\left(b^z\right)^t \ne b^{zt},</math>
unless {{mvar|z}} is real or {{mvar|t}} is an integer.

[[Euler's formula]],
: <math>e^{iy} = \cos y + i \sin y,</math>
allows expressing the polar form of <math>b^z</math> in terms of the real and imaginary parts of {{mvar|z}}, namely
: <math>b^{x+iy}= b^x(\cos(y\ln b)+i\sin(y\ln b)),</math>
where the [[Absolute value|absolute value]] of the [[Trigonometry|trigonometric]] factor is one. This results from
: <math>b^{x+iy}=b^x b^{iy}=b^x e^{iy\ln b} =b^x(\cos(y\ln b)+i\sin(y\ln b)).</math>

==Non-integer exponents with a complex base==
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 {{mvar|n}}th roots, that is, of exponents <math>1/n,</math> where {{mvar|n}} is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to {{mvar|n}}th roots, this case deserves to be considered first, since it does not need to use [[Complex logarithm|complex logarithm]]s, and is therefore easier to understand.

==={{mvar|n}}th roots of a complex number===
Every nonzero complex number {{mvar|z}} may be written in polar form as
:<math>z=\rho e^{i\theta}=\rho(\cos \theta +i \sin \theta),</math>
where <math>\rho</math> is the [[Absolute value|absolute value]] of {{mvar|z}}, and <math>\theta</math> is its [[Argument (complex analysis)|argument]]. The argument is defined [[Up to|up to]] an integer multiple of {{math|2{{pi}}}}; this means that, if <math>\theta</math> is the argument of a complex number, then <math>\theta +2k\pi</math> is also an argument of the same complex number for every integer <math>k</math>.

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 {{mvar|n}}th root of a complex number can be obtained by taking the {{mvar|n}}th root of the absolute value and dividing its argument by {{mvar|n}}:
: <math>\left(\rho e^{i\theta}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i\theta}n.</math>

If <math>2\pi</math> is added to <math>\theta</math>, the complex number is not changed, but this adds <math>2i\pi/n</math> to the argument of the {{mvar|n}}th root, and provides a new {{mvar|n}}th root. This can be done {{mvar|n}} times (<math>k=0,1,...,n-1</math>), and provides the {{mvar|n}} {{mvar|n}}th roots of the complex number:
: <math>\left(\rho e^{i(\theta+2k\pi)}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i(\theta+2k\pi)}n.</math>

It is usual to choose one of the {{mvar|n}} {{mvar|n}}th root as the principal root. The common choice is to choose the {{mvar|n}}th root for which <math>-\pi<\theta\le \pi,</math> that is, the {{mvar|n}}th root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal {{mvar|n}}th root a [[Continuous function|continuous function]] in the whole complex plane, except for negative real values of the radicand. This function equals the usual {{mvar|n}}th root for positive real radicands. For negative real radicands, and odd exponents, the principal {{mvar|n}}th root is not real, although the usual {{mvar|n}}th root is real. [[Analytic continuation]] shows that the principal {{mvar|n}}th root is the unique complex differentiable function that extends the usual {{mvar|n}}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 <math>2\pi,</math> the complex number comes back to its initial position, and its {{mvar|n}}th roots are permuted circularly (they are multiplied by <math DISPLAY=textstyle>e^{2i\pi/n}</math>). This shows that it is not possible to define a {{mvar|n}}th root function that is continuous in the whole complex plane.

====Roots of unity====
{{Main|Root of unity}}

[[File:One3Root.svg|thumb|right|The three third roots of {{math|1}}]]
The {{mvar|n}}th roots of unity are the {{mvar|n}} complex numbers such that {{math|1=''w''<sup>''n''</sup> = 1}}, where {{mvar|n}} is a positive integer. They arise in various areas of mathematics, such as in [[Discrete Fourier transform|discrete Fourier transform]] or algebraic solutions of algebraic equations (Lagrange resolvent).

The {{mvar|n}} {{mvar|n}}th roots of unity are the {{mvar|n}} first powers of <math>\omega =e^\frac{2\pi i}{n}</math>, that is <math>1=\omega^0=\omega^n, \omega=\omega^1, \omega^2,..., \omega^{n-1}.</math> The {{mvar|n}}th roots of unity that have this generating property are called ''primitive {{mvar|n}}th roots of unity''; they have the form <math>\omega^k=e^\frac{2k\pi i}{n},</math> with {{mvar|k}} [[Coprime integers|coprime]] with {{mvar|n}}. The unique primitive square root of unity is <math>-1;</math> the primitive fourth roots of unity are <math>i</math> and <math>-i.</math>

The {{mvar|n}}th roots of unity allow expressing all {{mvar|n}}th roots of a complex number {{mvar|z}} as the {{mvar|n}} products of a given {{mvar|n}}th roots of {{mvar|z}} with a {{mvar|n}}th root of unity.

Geometrically, the {{mvar|n}}th roots of unity lie on the [[Unit circle|unit circle]] of the [[Complex plane|complex plane]] at the vertices of a [[regular polygon|regular {{mvar|n}}-gon]] with one vertex on the real number 1.

As the number <math>e^\frac{2k\pi i}{n}</math> is the primitive {{mvar|n}}th root of unity with the smallest positive [[Argument (complex analysis)|argument]], it is called the ''principal primitive {{mvar|n}}th root of unity'', sometimes shortened as ''principal {{mvar|n}}th root of unity'', although this terminology can be confused with the [[Principal value|principal value]] of <math>1^{1/n}</math>, which is 1.<ref>{{cite book |title=Introduction to Algorithms |edition=second |author-last1=Cormen |author-first1=Thomas H. |author-last2=Leiserson |author-first2=Charles E. |author-last3=Rivest |author-first3=Ronald L. |author-last4=Stein |author-first4=Clifford |publisher=MIT Press |date=2001 |isbn=978-0-262-03293-3}} [http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html Online resource] {{webarchive|url=https://web.archive.org/web/20070930201902/http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html |date=2007-09-30}}.</ref><ref>{{cite book |title=Difference Equations: From Rabbits to Chaos |title-link= Difference Equations: From Rabbits to Chaos |edition=[[Undergraduate Texts in Mathematics]] |author-last1=Cull |author-first1=Paul |author-last2=Flahive |author-first2=Mary |author-last3=Robson |author-first3=Robby |date=2005 |publisher=Springer |isbn=978-0-387-23234-8}} Defined on p. 351.</ref><ref>{{MathWorld |title=Principal root of unity |id=PrincipalRootofUnity}}</ref>

===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 <math DISPLAY=textstyle>z^w</math>. So, either a [[Principal value|principal value]] is defined, which is not continuous for the values of {{mvar|z}} that are real and nonpositive, or <math DISPLAY=textstyle>z^w</math> is defined as a [[Multivalued function|multivalued function]].

In all cases, the [[Complex logarithm|complex logarithm]] is used to define complex exponentiation as
: <math>z^w=e^{w\log z},</math>
where <math>\log z</math> is the variant of the complex logarithm that is used, which is a function or a [[Multivalued function|multivalued function]] such that
: <math>e^{\log z}=z</math>
for every {{mvar|z}} in its [[Domain of a function|domain of definition]].

====Principal value====
The [[Principal value|principal value]] of the [[Complex logarithm|complex logarithm]] is the unique continuous function, commonly denoted <math>\log,</math> such that, for every nonzero complex number {{mvar|z}},
: <math>e^{\log z}=z,</math>
and the [[Argument (complex analysis)|argument]] of {{mvar|z}} satisfies
: <math>-\pi <\operatorname{Arg}z \le \pi.</math>
The principal value of the complex logarithm is not defined for <math>z=0,</math> it is [[Continuous function|discontinuous]] at negative real values of {{mvar|z}}, and it is holomorphic (that is, complex differentiable) elsewhere. If {{mvar|z}} is real and positive, the principal value of the complex logarithm is the natural logarithm: <math>\log z=\ln z.</math>

The principal value of <math>z^w</math> is defined as
<math>z^w=e^{w\log z},</math>
where <math>\log z</math> is the principal value of the logarithm.

The function <math>(z,w)\to z^w</math> is holomorphic except in the neighbourhood of the points where {{mvar|z}} is real and nonpositive.

If {{mvar|z}} is real and positive, the principal value of <math>z^w</math> equals its usual value defined above. If <math>w=1/n,</math> where {{mvar|n}} 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 <math>\log z</math> and <math>z^w</math> at the negative real values of {{mvar|z}}. In this case, it is useful to consider these functions as [[Multivalued function|multivalued function]]s.

If <math>\log z</math> denotes one of the values of the multivalued logarithm (typically its principal value), the other values are <math>2ik\pi +\log z,</math> where {{mvar|k}} is any integer. Similarly, if <math>z^w</math> is one value of the exponentiation, then the other values are given by
: <math>e^{w(2ik\pi +\log z)} = z^we^{2ik\pi w},</math>
where {{mvar|k}} is any integer.

Different values of {{mvar|k}} give different values of <math>z^w</math> unless {{mvar|w}} is a [[Rational number|rational number]], that is, there is an integer {{mvar|d}} such that {{mvar|dw}} is an integer. This results from the [[Periodic function|periodicity]] of the exponential function, more specifically, that <math>e^a=e^b</math> if and only if <math>a-b</math> is an integer multiple of <math>2\pi i.</math>

If <math>w=\frac mn</math> is a rational number with {{mvar|m}} and {{mvar|n}} [[Coprime integers|coprime integers]] with <math>n>0,</math> then <math>z^w</math> has exactly {{mvar|n}} values. In the case <math>m=1,</math> these values are the same as those described in [[#nth roots of a complex number|§ {{mvar|n}}th roots of a complex number]]. If {{mvar|w}} is an integer, there is only one value that agrees with that of {{slink||Integer exponents}}.

The multivalued exponentiation is holomorphic for <math>z\ne 0,</math> in the sense that its [[Graph of a function|graph]] consists of several sheets that define each a holomorphic function in the neighborhood of every point. If {{mvar|z}} varies continuously along a circle around {{math|0}}, then, after a turn, the value of <math>z^w</math> has changed of sheet.

====Computation====
The ''canonical form'' <math>x+iy</math> of <math>z^w</math> can be computed from the canonical form of {{mvar|z}} and {{mvar|w}}. Although this can be described by a single formula, it is clearer to split the computation in several steps.
* ''Polar form of {{mvar|z}}''. If <math>z=a+ib</math> is the canonical form of {{mvar|z}} ({{mvar|a}} and {{mvar|b}} being real), then its polar form is <math display=block>z=\rho e^{i\theta}= \rho (\cos\theta + i \sin\theta),</math> with <math display=inline>\rho=\sqrt{a^2+b^2}</math> and <math>\theta=\operatorname{atan2}(b,a)</math>, where {{tmath|\operatorname{atan2} }} is the two-argument arctangent function.
* ''[[Complex logarithm|Logarithm]] of {{mvar|z}}''. The [[Principal value|principal value]] of this logarithm is <math>\log z=\ln \rho+i\theta,</math> where <math>\ln</math> denotes the [[Natural logarithm|natural logarithm]]. The other values of the logarithm are obtained by adding <math>2ik\pi</math> for any integer {{mvar|k}}.
* ''Canonical form of <math>w\log z.</math>'' If <math>w=c+di</math> with {{mvar|c}} and {{mvar|d}} real, the values of <math>w\log z</math> are <math display=block>w\log z = (c\ln \rho - d\theta-2dk\pi) +i (d\ln \rho + c\theta+2ck\pi),</math> the principal value corresponding to <math>k=0.</math>
* ''Final result''. Using the identities <math>e^{x+y}=e^xe^y</math> and <math>e^{y\ln x} = x^y,</math> one gets <math DISPLAY=block>z^w=\rho^c e^{-d(\theta+2k\pi)} \left(\cos (d\ln \rho + c\theta+2ck\pi) +i\sin(d\ln \rho + c\theta+2ck\pi)\right),</math> with <math>k=0</math> for the principal value.

=====Examples=====
* <math>i^i</math> <br>The polar form of {{mvar|i}} is <math>i=e^{i\pi/2},</math> and the values of <math>\log i</math> are thus <math DISPLAY=block>\log i=i\left(\frac \pi 2 +2k\pi\right).</math> It follows that <math DISPLAY=block>i^i=e^{i\log i}=e^{-\frac \pi 2} e^{-2k\pi}.</math>So, all values of <math>i^i</math> are real, the principal one being <math DISPLAY=block> e^{-\frac \pi 2} \approx 0.2079.</math>
* <math>(-2)^{3+4i}</math><br>Similarly, the polar form of {{math|−2}} is <math>-2 = 2e^{i \pi}.</math> So, the above described method gives the values <math DISPLAY=block>\begin{align}
(-2)^{3 + 4i} &= 2^3 e^{-4(\pi+2k\pi)} (\cos(4\ln 2 + 3(\pi +2k\pi)) +i\sin(4\ln 2 + 3(\pi+2k\pi)))\\
&=-2^3 e^{-4(\pi+2k\pi)}(\cos(4\ln 2) +i\sin(4\ln 2)).
\end{align}</math>In this case, all the values have the same argument <math>4\ln 2,</math> and different absolute values.

In both examples, all values of <math>z^w</math> have the same argument. More generally, this is true if and only if the real part of {{mvar|w}} 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:

{{bulleted list
| The identity {{math|1=log(''b''<sup>''x''</sup>) = ''x'' ⋅ log ''b''}} holds whenever {{mvar|b}} is a positive real number and {{mvar|x}} is a real number. But for the [[Principal branch|principal branch]] of the complex logarithm one has
<math display="block"> \log((-i)^2) = \log(-1) = i\pi \neq 2\log(-i) = 2\log(e^{-i\pi/2})=2\,\frac{-i\pi}{2} = -i\pi</math>

Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:
<math display="block">\log w^z \equiv z \log w \pmod{2 \pi i}</math>

This identity does not hold even when considering log as a multivalued function. The possible values of {{math|log(''w''<sup>''z''</sup>)}} contain those of {{math|''z'' ⋅ log ''w''}} as a proper subset. Using {{math|Log(''w'')}} for the principal value of {{math|log(''w'')}} and {{mvar|m}}, {{mvar|n}} as any integers the possible values of both sides are:
<math display="block">\begin{align}
      \left\{\log w^z \right\} &= \left\{ z \cdot \operatorname{Log} w + z \cdot 2 \pi i n + 2 \pi i m \mid m,n\in\Z\right\} \\
      \left\{z \log w \right\} &= \left\{ z \operatorname{Log} w + z \cdot 2 \pi i n \mid n\in \Z\right\}
\end{align}</math>
| The identities {{math|1=(''bc'')<sup>''x''</sup> = ''b''<sup>''x''</sup>''c''<sup>''x''</sup>}} and {{math|1=(''b''/''c'')<sup>''x''</sup> = ''b''<sup>''x''</sup>/''c''<sup>''x''</sup>}} are valid when {{mvar|b}} and {{mvar|c}} are positive real numbers and {{mvar|x}} is a real number. But, for the principal values, one has
<math display="block">(-1 \cdot -1)^\frac{1}{2} =1 \neq (-1)^\frac{1}{2} (-1)^\frac{1}{2} =i \cdot i=i^2 =-1</math>
and
<math display="block">\left(\frac{1}{-1}\right)^\frac{1}{2} = (-1)^\frac{1}{2} = i \neq \frac{1^\frac{1}{2}}{(-1)^\frac{1}{2}} = \frac{1}{i} = -i</math>

On the other hand, when {{mvar|x}} is an integer, the identities are valid for all nonzero complex numbers.

If exponentiation is considered as a multivalued function then the possible values of {{math|(−1 ⋅ −1)<sup>1/2</sup>}} are {{math|{{mset|1, −1}}}}. The identity holds, but saying {{math|1={1} = {{mset|(−1 ⋅ −1)<sup>1/2</sup>}}}} is incorrect.

For any integer {{mvar|n}}, we have:
# <math>e^{1 + 2 \pi i n} = e^1 e^{2 \pi i n} = e \cdot 1 = e</math>
# <math>\left(e^{1 + 2\pi i n}\right)^{1 + 2 \pi i n} = e\qquad</math> (taking the <math>(1 + 2 \pi i n)</math>-th power of both sides)
# <math>e^{1 + 4 \pi i n - 4 \pi^2 n^2} = e\qquad</math> (using <math>\left(e^x\right)^y = e^{xy}</math> and expanding the exponent)
# <math>e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e\qquad</math> (using <math>e^{x+y} = e^x e^y</math>)
# <math>e^{-4 \pi^2 n^2} = 1\qquad</math> (dividing by {{mvar|e}})

but this is false when the integer {{mvar|n}} is nonzero.

The error is the following: by definition, <math>e^y</math> is a notation for <math>\exp(y),</math> a true function, and <math>x^y</math> is a notation for <math>\exp(y\log x),</math> which is a multi-valued function. Thus the notation is ambiguous when {{math|1=''x'' = ''e''}}. Here, before expanding the exponent, the second line should be
<math display="block">\exp\left((1 + 2\pi i n)\log \exp(1 + 2\pi i n)\right) = \exp(1 + 2\pi i n).</math>

Therefore, when expanding the exponent, one has implicitly supposed that <math>\log \exp z =z</math> for complex values of {{mvar|z}}, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity {{math|1=(''e''<sup>''x''</sup>)<sup>''y''</sup> = ''e''<sup>''xy''</sup>}} must be replaced by the identity
<math display="block">\left(e^x\right)^y = e^{y\log e^x},</math>
which is a true identity between multivalued functions.
}}

==Irrationality and transcendence==
{{Main|Gelfond–Schneider theorem}}
If {{mvar|b}} is a positive real [[Algebraic number|algebraic number]], and {{mvar|x}} is a rational number, then {{math|''b''<sup>''x''</sup>}} is an algebraic number. This results from the theory of [[Algebraic extension|algebraic extension]]s. This remains true if {{mvar|b}} is any algebraic number, in which case, all values of {{math|''b''<sup>''x''</sup>}} (as a [[Multivalued function|multivalued function]]) are algebraic. If {{mvar|x}} is [[Irrational number|irrational]] (that is, ''not rational''), and both {{mvar|b}} and {{mvar|x}} are algebraic, Gelfond–Schneider theorem asserts that all values of {{math|''b''<sup>''x''</sup>}} are [[Transcendental number|transcendental]] (that is, not algebraic), except if {{mvar|b}} equals {{math|0}} or {{math|1}}.

In other words, if {{mvar|x}} is irrational and <math>b\not\in \{0,1\},</math> then at least one of {{mvar|b}}, {{mvar|x}} and {{math|''b''<sup>''x''</sup>}} 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.<ref group="nb">More generally, [[Power associativity|power associativity]] is sufficient for the definition.</ref> The definition of {{math|''x''<sup>0</sup>}} requires further the existence of a multiplicative identity.<ref>{{cite book |author-last=Bourbaki |author-first=Nicolas |title=Algèbre |date=1970 |publisher=Springer|at=I.2}}</ref>

An [[Algebraic structure|algebraic structure]] consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by {{math|1}} is a [[Monoid|monoid]]. In such a monoid, exponentiation of an element {{mvar|x}} is defined inductively by
* <math>x^0 = 1,</math>
* <math>x^{n+1} = x x^n</math> for every nonnegative integer {{mvar|n}}.

If {{mvar|n}} is a negative integer, <math>x^n</math> is defined only if {{mvar|x}} has a [[Multiplicative inverse|multiplicative inverse]].<ref>{{cite book |author-last=Bloom |author-first=David M. |title=Linear Algebra and Geometry |date=1979 |isbn=978-0-521-29324-2 |page=[https://archive.org/details/linearalgebrageo0000bloo/page/45 45] |publisher=Cambridge University Press |url=https://archive.org/details/linearalgebrageo0000bloo |url-access=registration}}</ref> In this case, the inverse of {{mvar|x}} is denoted {{math|''x''<sup>−1</sup>}}, and {{math|''x''<sup>''n''</sup>}} is defined as <math>\left(x^{-1}\right)^{-n}.</math>

Exponentiation with integer exponents obeys the following laws, for {{mvar|x}} and {{mvar|y}} in the algebraic structure, and {{mvar|m}} and {{mvar|n}} integers:
: <math>\begin{align}
x^0&=1\\
x^{m+n}&=x^m x^n\\
(x^m)^n&=x^{mn}\\
(xy)^n&=x^n y^n \quad \text{if } xy=yx, \text{and, in particular, if the multiplication is commutative.}
\end{align}</math>

These definitions are widely used in many areas of mathematics, notably for [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], [[Field (mathematics)|fields]], [[Square matrix|square matrices]] (which form a ring). They apply also to [[Function (mathematics)|functions]] from a [[Set (mathematics)|set]] to itself, which form a monoid under [[Function composition|function composition]]. This includes, as specific instances, [[Geometric transformation|geometric transformation]]s, and [[Endomorphism|endomorphism]]s of any [[Mathematical structure|mathematical structure]].

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if {{mvar|f}} is a [[Real function|real function]] whose valued can be multiplied, <math>f^n</math> denotes the exponentiation with respect of multiplication, and <math>f^{\circ n}</math> may denote exponentiation with respect of [[Function composition|function composition]]. That is,
: <math>(f^n)(x)=(f(x))^n=f(x) \,f(x) \cdots f(x),</math>
and
: <math>(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots)).</math>
Commonly, <math>(f^n)(x)</math> is denoted <math>f(x)^n,</math> while <math>(f^{\circ n})(x)</math> is denoted <math>f^n(x).</math>

===In a group===
A [[Multiplicative group|multiplicative group]] is a set with as associative operation denoted as multiplication, that has an [[Identity element|identity element]], and such that every element has an inverse.

So, if {{mvar|G}} is a group, <math>x^n</math> is defined for every <math>x\in G</math> and every integer {{mvar|n}}.

The set of all powers of an element of a group form a [[Subgroup|subgroup]]. A group (or subgroup) that consists of all powers of a specific element {{mvar|x}} is the [[Cyclic group|cyclic group]] generated by {{mvar|x}}. If all the powers of {{mvar|x}} are distinct, the group is isomorphic to the [[Additive group|additive group]] <math>\Z</math> of the integers. Otherwise, the cyclic group is [[Finite group|finite]] (it has a finite number of elements), and its number of elements is the [[Order (group theory)|order]] of {{mvar|x}}. If the order of {{mvar|x}} is {{mvar|n}}, then <math>x^n=x^0=1,</math> and the cyclic group generated by {{mvar|x}} consists of the {{mvar|n}} first powers of {{mvar|x}} (starting indifferently from the exponent {{math|0}} or {{math|1}}).

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

Superscript notation is also used for [[Conjugacy class|conjugation]]; that is, {{math|1=''g''<sup>''h''</sup> = ''h''<sup>−1</sup>''gh''}}, where {{math|''g''}} and {{math|''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 <math>(g^h)^k=g^{hk}</math> and <math>(gh)^k=g^kh^k.</math>

===In a ring===
In a [[Ring (mathematics)|ring]], it may occur that some nonzero elements satisfy <math>x^n=0</math> for some integer {{mvar|n}}. Such an element is said to be [[Nilpotent|nilpotent]]. In a [[Commutative ring|commutative ring]], the nilpotent elements form an [[Ideal (ring theory)|ideal]], called the [[Nilradical of a ring|nilradical]] of the ring.

If the nilradical is reduced to the zero ideal (that is, if <math>x\neq 0</math> implies <math>x^n\neq 0</math> for every positive integer {{mvar|n}}), the commutative ring is said to be [[Reduced ring|reduced]]. Reduced rings are important in [[Algebraic geometry|algebraic geometry]], since the coordinate ring of an [[Affine algebraic set|affine algebraic set]] is always a reduced ring.

More generally, given an ideal {{mvar|I}} in a commutative ring {{mvar|R}}, the set of the elements of {{mvar|R}} that have a power in {{mvar|I}} is an ideal, called the [[Radical of an ideal|radical]] of {{mvar|I}}. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a [[Polynomial ring|polynomial ring]] <math>k[x_1, \ldots, x_n]</math> over a [[Field (mathematics)|field]] {{mvar|k}}, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of [[Hilbert's Nullstellensatz]]).

===Matrices and linear operators===
If {{math|''A''}} is a square matrix, then the product of {{math|''A''}} with itself {{math|''n''}} times is called the matrix power. Also <math>A^0</math> is defined to be the identity matrix,<ref>Chapter 1, Elementary Linear Algebra, 8E, Howard Anton.</ref> and if {{math|''A''}} is invertible, then <math>A^{-n} = \left(A^{-1}\right)^n</math>.

Matrix powers appear often in the context of discrete dynamical systems, where the matrix {{math|''A''}} expresses a transition from a state vector {{math|''x''}} of some system to the next state {{math|''Ax''}} of the system.<ref>{{citation |last=Strang |first=Gilbert |title=Linear algebra and its applications |publisher=Brooks-Cole |date=1988 |edition=3rd|at=Chapter 5}}</ref> This is the standard interpretation of a [[Markov chain]], for example. Then <math>A^2x</math> is the state of the system after two time steps, and so forth: <math>A^nx</math> is the state of the system after {{math|''n''}} time steps. The matrix power <math>A^n</math> is the transition matrix between the state now and the state at a time {{math|''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|eigenvalues and eigenvectors]].

Apart from matrices, more general [[Linear operator|linear operator]]s can also be exponentiated. An example is the [[Derivative|derivative]] operator of calculus, <math>d/dx</math>, which is a linear operator acting on functions <math>f(x)</math> to give a new function <math>(d/dx)f(x) = f'(x)</math>. The {{math|''n''}}th power of the differentiation operator is the {{math|''n''}}th derivative:
: <math>\left(\frac{d}{dx}\right)^nf(x) = \frac{d^n}{dx^n}f(x) = f^{(n)}(x).</math>

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 [[C0-semigroup|semigroups]].<ref>E. Hille, R. S. Phillips: ''Functional Analysis and Semi-Groups''. American Mathematical Society, 1975.</ref> 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|heat equation]], [[Schrödinger equation]], [[Wave equation|wave equation]], and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the [[Fractional calculus|fractional calculus]].

===Finite fields===
{{Main|Finite field}}
A [[Field (mathematics)|field]] is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a [[Multiplicative inverse|multiplicative inverse]]. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of {{math|0}}. Common examples are the field of [[Complex number|complex number]]s, the [[Real number|real number]]s and the [[Rational number|rational number]]s, considered earlier in this article, which are all [[Infinite set|infinite]].

A ''finite field'' is a field with a [[Finite set|finite number]] of elements. This number of elements is either a [[Prime number|prime number]] or a [[Prime power|prime power]]; that is, it has the form <math>q=p^k,</math> where {{mvar|p}} is a prime number, and {{mvar|k}} is a positive integer. For every such {{mvar|q}}, there are fields with {{mvar|q}} elements. The fields with {{mvar|q}} elements are all isomorphic, which allows, in general, working as if there were only one field with {{mvar|q}} elements, denoted <math>\mathbb F_q.</math>

One has
: <math>x^q=x</math>
for every <math>x\in \mathbb F_q.</math>

A [[Primitive element (finite field)|primitive element]] in <math>\mathbb F_q</math> is an element {{mvar|g}} such that the set of the {{math|''q'' − 1}} first powers of {{mvar|g}} (that is, <math>\{g^1=g, g^2, \ldots, g^{p-1}=g^0=1\}</math>) equals the set of the nonzero elements of <math>\mathbb F_q.</math> There are <math>\varphi (p-1)</math> primitive elements in <math>\mathbb F_q,</math> where <math>\varphi</math> is [[Euler's totient function]].

In <math>\mathbb F_q,</math> the [[Freshman's dream|freshman's dream]] identity
: <math>(x+y)^p = x^p+y^p</math>
is true for the exponent {{mvar|p}}. As <math>x^p=x</math> in <math>\mathbb F_q,</math> It follows that the map
: <math>\begin{align}
F\colon{} & \mathbb F_q \to \mathbb F_q\\
& x\mapsto x^p
\end{align}</math>
is [[Linear map|linear]] over <math>\mathbb F_q,</math> and is a field automorphism, called the [[Frobenius automorphism]]. If <math>q=p^k,</math> the field <math>\mathbb F_q</math> has {{mvar|k}} automorphisms, which are the {{mvar|k}} first powers (under [[Function composition|composition]]) of {{mvar|F}}. In other words, the [[Galois group]] of <math>\mathbb F_q</math> is [[Cyclic group|cyclic]] of order {{mvar|k}}, generated by the Frobenius automorphism.

The [[Diffie–Hellman key exchange]] is an application of exponentiation in finite fields that is widely used for [[Secure communication|secure communication]]s. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the [[Discrete logarithm|discrete logarithm]], is computationally expensive. More precisely, if {{mvar|g}} is a primitive element in <math>\mathbb F_q,</math> then <math>g^e</math> can be efficiently computed with [[Exponentiation by squaring|exponentiation by squaring]] for any {{mvar|e}}, even if {{mvar|q}} is large, while there is no known computationally practical algorithm that allows retrieving {{mvar|e}} from <math>g^e</math> if {{mvar|q}} is sufficiently large.

==Powers of sets {{Anchor|Exponentiation over sets}}==
The [[Cartesian product]] of two [[Set (mathematics)|sets]] {{mvar|S}} and {{mvar|T}} is the set of the [[Ordered pair|ordered pair]]s <math>(x,y)</math> such that <math>x\in S</math> and <math>y\in T.</math> This operation is not properly commutative nor associative, but has these properties [[Up to|up to]] [[Canonical map|canonical]] [[Isomorphism|isomorphism]]s, that allow identifying, for example, <math>(x,(y,z)),</math> <math>((x,y),z),</math> and <math>(x,y,z).</math>

This allows defining the {{mvar|n}}th power <math>S^n</math> of a set {{mvar|S}} as the set of all {{mvar|n}}-[[Tuple|tuple]]s <math>(x_1, \ldots, x_n)</math> of elements of {{mvar|S}}.

When {{mvar|S}} is endowed with some structure, it is frequent that <math>S^n</math> is naturally endowed with a similar structure. In this case, the term "[[Direct product|direct product]]" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example <math>\R^n</math> (where <math>\R</math> denotes the real numbers) denotes the Cartesian product of {{mvar|n}} copies of <math>\R,</math> as well as their direct product as [[Vector space|vector space]], [[Topological space|topological space]]s, [[Ring (mathematics)|rings]], etc.

===Sets as exponents===
A {{mvar|n}}-tuple <math>(x_1, \ldots, x_n)</math> of elements of {{mvar|S}} can be considered as a [[Function (mathematics)|function]] from <math>\{1,\ldots, n\}.</math> This generalizes to the following notation.

Given two sets {{mvar|S}} and {{mvar|T}}, the set of all functions from {{mvar|T}} to {{mvar|S}} is denoted <math>S^T</math>. This exponential notation is justified by the following canonical isomorphisms (for the first one, see [[Currying]]):
: <math>(S^T)^U\cong S^{T\times U},</math>
: <math>S^{T\sqcup U}\cong S^T\times S^U,</math>
where <math>\times</math> denotes the Cartesian product, and <math>\sqcup</math> the [[Disjoint union|disjoint union]].

One can use sets as exponents for other operations on sets, typically for [[Direct sum|direct sum]]s of [[Abelian group|abelian group]]s, [[Vector space|vector space]]s, or [[Module (mathematics)|modules]]. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, <math>\R^\N</math> denotes the vector space of the infinite sequences of real numbers, and <math>\R^{(\N)}</math> the vector space of those sequences that have a finite number of nonzero elements. The latter has a [[Basis (linear algebra)|basis]] consisting of the sequences with exactly one nonzero element that equals {{math|1}}, while the Hamel bases of the former cannot be explicitly described (because their existence involves [[Zorn's lemma]]).

In this context, {{math|2}} can represents the set <math>\{0,1\}.</math> So, <math>2^S</math> denotes the [[Power set|power set]] of {{mvar|S}}, that is the set of the functions from {{mvar|S}} to <math>\{0,1\},</math> which can be identified with the set of the [[Subset|subset]]s of {{mvar|S}}, by mapping each function to the inverse image of {{math|1}}.

This fits in with the exponentiation of cardinal numbers, in the sense that {{math|1={{abs|''S''<sup>''T''</sup>}} = {{abs|''S''}}<sup>{{abs|''T''}}</sup>}}, where {{math|{{abs|''X''}}}} is the cardinality of {{math|''X''}}.

===In category theory===
{{Main|Cartesian closed category}}
In the [[Category of sets|category of sets]], the [[Morphism|morphism]]s between sets {{mvar|X}} and {{mvar|Y}} are the functions from {{mvar|X}} to {{mvar|Y}}. It results that the set of the functions from {{mvar|X}} to {{mvar|Y}} that is denoted <math>Y^X</math> in the preceding section can also be denoted <math>\hom(X,Y).</math> The isomorphism <math>(S^T)^U\cong S^{T\times U}</math> can be rewritten
:<math>\hom(U,S^T)\cong \hom(T\times U,S).</math>
This means the functor "exponentiation to the power {{mvar|T{{space|thin}}}}" is a right adjoint to the functor "direct product with {{mvar|T{{space|thin}}}}".

This generalizes to the definition of exponentiation in a category in which finite [[Direct product|direct product]]s exist: in such a category, the functor <math>X\to X^T</math> is, if it exists, a right adjoint to the functor <math>Y\to T\times Y.</math> A category is called a ''Cartesian closed category'', if direct products exist, and the functor <math>Y\to X\times Y</math> has a right adjoint for every {{mvar|T}}.

==Repeated exponentiation==
{{Main|Tetration|Hyperoperation}}
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. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the [[Ackermann function]] 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 {{math|(3, 3)}}, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and {{val|7625597484987}} ({{math|1==3<sup>27</sup> = 3<sup>3<sup>3</sup></sup> = <sup>3</sup>3}}) respectively.

==Limits of powers==
[[Zero to the power of zero]] gives a number of examples of limits that are of the [[Indeterminate form|indeterminate form]] 0<sup>0</sup>. The limits in these examples exist, but have different values, showing that the two-variable function {{math|''x''<sup>''y''</sup>}} has no limit at the point {{math|(0, 0)}}. One may consider at what points this function does have a limit.

More precisely, consider the function <math>f(x,y) = x^y</math> defined on <math> D = \{(x, y) \in \mathbf{R}^2 : x > 0 \}</math>. Then {{math|''D''}} can be viewed as a subset of {{math|{{overline|'''R'''}}<sup>2</sup>}} (that is, the set of all pairs {{math|(''x'', ''y'')}} with {{math|''x''}}, {{math|''y''}} belonging to the [[Extended real number line|extended real number line]] {{math|1={{overline|'''R'''}} = [−∞, +∞]}}, endowed with the [[Product topology|product topology]]), which will contain the points at which the function {{math|''f''}} has a limit.

In fact, {{math|''f''}} has a limit at all [[Accumulation point|accumulation point]]s of {{math|''D''}}, except for {{math|(0, 0)}}, {{math|(+∞, 0)}}, {{math|(1, +∞)}} and {{math|(1, −∞)}}.<ref>Nicolas Bourbaki, ''Topologie générale'', V.4.2.</ref> Accordingly, this allows one to define the powers {{math|''x''<sup>''y''</sup>}} by continuity whenever {{math|0 ≤ ''x'' ≤ +∞}}, {{math|−∞ ≤ y ≤ +∞}}, except for {{math|0<sup>0</sup>}}, {{math|(+∞)<sup>0</sup>}}, {{math|1<sup>+∞</sup>}} and {{math|1<sup>−∞</sup>}}, which remain indeterminate forms.

Under this definition by continuity, we obtain:
* {{math|1=''x''<sup>+∞</sup> = +∞}} and {{math|1=''x''<sup>−∞</sup> = 0}}, when {{math|1 < ''x'' ≤ +∞}}.
* {{math|1=''x''<sup>+∞</sup> = 0}} and {{math|1=''x''<sup>−∞</sup> = +∞}}, when {{math|0 < ''x'' < 1}}.
* {{math|1=0<sup>''y''</sup> = 0}} and {{math|1=(+∞)<sup>''y''</sup> = +∞}}, when {{math|0 < ''y'' ≤ +∞}}.
* {{math|1=0<sup>''y''</sup> = +∞}} and {{math|1=(+∞)<sup>''y''</sup> = 0}}, when {{math|−∞ ≤ ''y'' < 0}}.

These powers are obtained by taking limits of {{math|''x''<sup>''y''</sup>}} for ''positive'' values of {{math|''x''}}. This method does not permit a definition of {{math|''x''<sup>''y''</sup>}} when {{math|''x'' < 0}}, since pairs {{math|(''x'', ''y'')}} with {{math|''x'' < 0}} are not accumulation points of {{math|''D''}}.

On the other hand, when {{math|''n''}} is an integer, the power {{math|''x''<sup>''n''</sup>}} is already meaningful for all values of {{math|''x''}}, including negative ones. This may make the definition {{math|1=0<sup>''n''</sup> = +∞}} obtained above for negative {{math|''n''}} problematic when {{math|''n''}} is odd, since in this case {{math|''x''<sup>''n''</sup> → +∞}} as {{math|''x''}} tends to {{math|0}} through positive values, but not negative ones.

==Efficient computation with integer exponents==
Computing {{math|''b''<sup>''n''</sup>}} using iterated multiplication requires {{math|''n'' − 1}} multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute {{math|2<sup>100</sup>}}, apply [[Horner's rule]] to the exponent 100 written in binary:
: <math>100 = 2^2 +2^5 + 2^6 = 2^2(1+2^3(1+2))</math>.
Then compute the following terms in order, reading Horner's rule from right to left.

{{Static row numbers}}
{| class="wikitable sortable static-row-numbers" style="text-align:right;"
|-
| 2<sup>2</sup> = 4
|-
| 2 (2<sup>2</sup>) = 2<sup>3</sup> = 8
|-
| (2<sup>3</sup>)<sup>2</sup> = 2<sup>6</sup> = 64
|-
| (2<sup>6</sup>)<sup>2</sup> = 2<sup>12</sup> = {{val|4,096}}
|-
| (2<sup>12</sup>)<sup>2</sup> = 2<sup>24</sup> = {{val|16,777,216}}
|-
| 2 (2<sup>24</sup>) = 2<sup>25</sup> = {{val|33,554,432}}
|-
| (2<sup>25</sup>)<sup>2</sup> = 2<sup>50</sup> = {{val|1,125,899,906,842,624}}
|-
| (2<sup>50</sup>)<sup>2</sup> = 2<sup>100</sup> = {{val|1,267,650,600,228,229,401,496,703,205,376}}
|}
This series of steps only requires 8 multiplications instead of 99.

In general, the number of multiplication operations required to compute {{math|''b''<sup>''n''</sup>}} can be reduced to <math>\sharp n +\lfloor \log_{2} n\rfloor -1,</math> by using [[Exponentiation by squaring|exponentiation by squaring]], where <math>\sharp n</math> denotes the number of {{math|1}}s in the binary representation of {{mvar|n}}. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal [[Addition-chain exponentiation|addition-chain exponentiation]]. Finding the ''minimal'' sequence of multiplications (the minimal-length addition chain for the exponent) for {{math|''b''<sup>''n''</sup>}} is a difficult problem, for which no efficient algorithms are currently known (see [[Subset sum problem]]), but many reasonably efficient heuristic algorithms are available.<ref>{{cite journal |last1=Gordon |first1=D. M. |title=A Survey of Fast Exponentiation Methods |journal=Journal of Algorithms |volume=27 |pages=129–146 |date=1998 |citeseerx=10.1.1.17.7076 |doi=10.1006/jagm.1997.0913 |url=http://www.ccrwest.org/gordon/jalg.pdf |access-date=2024-01-11  |archive-date=2018-07-23  |archive-url=https://web.archive.org/web/20180723164121/http://www.ccrwest.org/gordon/jalg.pdf |url-status=dead }}</ref> However, in practical computations, exponentiation by squaring is efficient enough, and much easier to implement.

==Iterated functions==
{{See also|Iterated function}}
[[Function composition]] is a [[Binary operation|binary operation]] that is defined on [[Function (mathematics)|functions]] such that the [[Codomain|codomain]] of the function written on the right is included in the [[Domain of a function|domain]] of the function written on the left. It is denoted <math>g\circ f,</math> and defined as
: <math>(g\circ f)(x)=g(f(x))</math>
for every {{mvar|x}} in the domain of {{mvar|f}}.

If the domain of a function {{mvar|f}} equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the {{mvar|n}}th power of the function under composition, commonly called the ''{{mvar|n}}th iterate'' of the function. Thus <math>f^n</math> denotes generally the {{mvar|n}}th iterate of {{mvar|f}}; for example, <math>f^3(x)</math> means <math>f(f(f(x))).</math><ref name="Peano_1903"/>

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration ''before'' the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication ''after'' the parentheses. Thus <math>f^2(x)= f(f(x)),</math> and <math>f(x)^2= f(x)\cdot f(x).</math> When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example <math>f^{\circ 3}=f\circ f \circ f,</math> and <math>f^3=f\cdot f\cdot f.</math> For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the [[Trigonometric functions|trigonometric functions]]. So, <math>\sin^2 x</math> and <math>\sin^2(x)</math> both mean <math>\sin(x)\cdot\sin(x)</math> and not <math>\sin(\sin(x)),</math> which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Cajori_1929"/>

In this context, the exponent <math>-1</math> denotes always the [[Inverse function|inverse function]], if it exists. So <math>\sin^{-1}x=\sin^{-1}(x) = \arcsin x.</math> For the [[Multiplicative inverse|multiplicative inverse]] fractions are generally used as in <math>1/\sin(x)=\frac 1{\sin x}.</math>

==In programming languages==
[[Programming language]]s generally express exponentiation either as an infix [[Operator (computer programming)|operator]] or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the [[Caret|caret]] (<code>^</code>). The [[ASCII#1963|original version of ASCII]] included an uparrow symbol (<code>↑</code>), intended for exponentiation, but this was [[Caret#History|replaced by the caret]] in 1967, so the caret became usual in programming languages.<ref>{{cite book|author=Richard Gillam|title=Unicode Demystified: A Practical Programmer's Guide to the Encoding Standard|year=2003|isbn=0201700522|page=33|publisher=Addison-Wesley Professional }}</ref>
The notations include:
* <code>x ^ y</code>: [[AWK]], [[BASIC]], J, [[Software:MATLAB|MATLAB]], [[Wolfram Language]] ([[Software:Wolfram Mathematica|Mathematica]]), [[R (programming language)|R]], [[Software:Microsoft Excel|Microsoft Excel]], [[Company:Analytica|Analytica]], [[Software:TeX|TeX]] (and its derivatives), [[TI-BASIC]], bc (for integer exponents), [[Haskell (programming language)|Haskell]] (for nonnegative integer exponents), [[Lua (programming language)|Lua]], and most [[Software:Computer algebra system|computer algebra system]]s.
* <code>x ** y</code>. The [[Fortran]] character set did not include lowercase characters or punctuation symbols other than <code>+-*/()&amp;=.,'</code> and so used <code>**</code> for exponentiation<ref name="Sayre_1956"/><ref>{{citation|author1=Brice Carnahan|author2=James O. Wilkes|title=Introduction to Digital Computing and FORTRAN IV with MTS Applications|year=1968|pages=2-2, 2-6}}</ref> (the initial version used <code>a xx b</code> instead.<ref name="Backus_1954"/>). Many other languages followed suit: [[Ada (programming language)|Ada]], [[Software:Z shell|Z shell]], [[Software:KornShell|KornShell]], [[Software:Bash (Unix shell)|Bash]], [[COBOL]], [[CoffeeScript]], [[Fortran]], FoxPro, [[Software:Gnuplot|Gnuplot]], [[Apache Groovy|Groovy]], [[JavaScript]], [[OCaml]], [[Object REXX|ooRexx]], [[F Sharp (programming language)|F#]], [[Perl]], [[PHP]], PL/I, [[Python (programming language)|Python]], [[Rexx]], [[Ruby (programming language)|Ruby]], SAS, [[Tcl]], [[ABAP]], [[Mercury (programming language)|Mercury]], Haskell (for floating-point exponents), [[Turing (programming language)|Turing]], and [[VHDL]].
* <code>x ↑ y</code>: Algol Reference language, [[Commodore BASIC]], TRS-80 Level II/III BASIC.<ref name="InfoWorld_1982">{{cite news |title=BASCOM - A BASIC compiler for TRS-80 I and II |author-last=Daneliuk |author-first=Timothy "Tim" A. |date=1982-08-09 |newspaper=InfoWorld |series=Software Reviews |publisher=Popular Computing, Inc. |volume=4 |number=31 |pages=41–42 |url=https://books.google.com/books?id=NDAEAAAAMBAJ&pg=PA42 |access-date=2020-02-06 |url-status=live |archive-url=https://web.archive.org/web/20200207104336/https://books.google.de/books?id=NDAEAAAAMBAJ&pg=PA42&focus=viewport#v=onepage&q=TRS-80%20exponention |archive-date=2020-02-07}}</ref><ref name="80Micro_1983">{{cite journal |title=80 Contents |journal=80 Micro |publisher=1001001, Inc. |issn=0744-7868 |date=October 1983 |issue=45 |page=5 |url=https://archive.org/details/80-microcomputing-magazine-1983-10 |access-date=2020-02-06}}</ref>
* <code>x ^^ y</code>: Haskell (for fractional base, integer exponents), [[D (programming language)|D]].
* <code>x⋆y</code>: [[APL (programming language)|APL]].

In most programming languages with an infix exponentiation operator, it is [[Operator associativity|right-associative]], that is, <code>a^b^c</code> is interpreted as <code>a^(b^c)</code>.<ref>{{cite book|author=Robert W. Sebesta|title=Concepts of Programming Languages|year=2010|isbn=978-0136073475|pages=130, 324|publisher=Addison-Wesley }}</ref> This is because <code>(a^b)^c</code> is equal to <code>a^(b*c)</code> and thus not as useful. In some languages, it is left-associative, notably in [[Algol]], [[Software:MATLAB|MATLAB]], and the Microsoft Excel formula language.

Other programming languages use functional notation:
* <code>(expt x y)</code>: [[Common Lisp]].
* <code>pown x y</code>: [[F Sharp (programming language)|F#]] (for integer base, integer exponent).

Still others only provide exponentiation as part of standard [[Library (computing)|libraries]]:
* <code>pow(x, y)</code>: [[C (programming language)|C]], [[C++]] (in <code>math</code> library).
* <code>Math.Pow(x, y)</code>: [[C Sharp (programming language)|C#]].
* <code>math:pow(X, Y)</code>: [[Erlang (programming language)|Erlang]].
* <code>Math.pow(x, y)</code>: [[Java (programming language)|Java]].
* <code>[Math]::Pow(x, y)</code>: [[PowerShell]].
In some [[Type system|statically typed]] languages that prioritize [[Type safety|type safety]] such as [[Rust (programming language)|Rust]], exponentiation is performed via a multitude of methods:

* <code>x.pow(y)</code> for <code>x</code> and <code>y</code> as integers
* <code>x.powf(y)</code> for <code>x</code> and <code>y</code> as floating-point numbers
* <code>x.powi(y)</code> for <code>x</code> as a float and <code>y</code> as an integer

==See also==
<!-- Please keep entries in alphabetical order & add a short description WP:SEEALSO -->
{{div col|colwidth=20em}}
==Notes==
{{Reflist|group="nb"|refs=
<!-- <ref group="nb" name="NB_Rucker">[[Biography:Alfred Pringsheim|Alfred Pringsheim]]'s and Jules Molk's (1907) notation {{math|{{i sup|''n''}}''f''(''x'')}} to denote [[Function composition|function composition]]s must not be confused with Rudolf von Bitter Rucker's (1982) notation {{math|{{i sup|''n''}}''x''}}, introduced by Hans Maurer (1901) and Reuben Louis Goodstein (1947) for tetration, or with David Patterson Ellerman's (1995) {{math|{{i sup|''n''}}''x''}} pre-superscript notation for [[Nth root|root]]s. --><!-- See {{cite book |title=Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics |chapter=Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics: Series Chauvinsism |series=G – Reference, Information and Interdisciplinary Subjects Series |work=The worldly philosophy: studies in intersection of philosophy and economics |author-last=Ellerman |author-first=David Patterson |edition=illustrated |publisher=Rowman & Littlefield Publishers, Inc. |date=1995-03-21 |isbn=0-8476-7932-2 |pages=237–268 [239] |chapter-url=https://books.google.com/books?id=NgJqXXk7zAAC&pg=PA237 |url=https://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20160305012729/http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |archive-date=2016-03-05}} [https://web.archive.org/web/20150917191423/http://www.ellerman.org/Davids-Stuff/Maths/sp_math.doc] (271 pages) --><!-- {{cite web |title=Introduction to Series-Parallel Duality |author-last=Ellerman |author-first=David Patterson |author-link=David Patterson Ellerman |publisher=University of California at Riverside |date=May 2004 |orig-year=1995-03-21 |citeseerx=10.1.1.90.3666 |url=https://www.ellerman.org/wp-content/uploads/2012/12/Series-Parallel-Duality.CV_.pdf |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20190810011716/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf |archive-date=2019-08-10}} [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf] (24 pages)</ref> -->
}}

==References==
{{Reflist|refs=
<ref name="Backus_1954">{{cite book |title=Specifications for: The IBM Mathematical FORmula TRANSlating System, FORTRAN |type=Preliminary report |author-last1=Backus |author-first1=John Warner |author-last2=Herrick |author-first2=Harlan L. |author-last3=Nelson |author-first3=Robert A. |author-last4=Ziller |author-first4=Irving |editor-last=Backus |editor-first=John Warner |publisher=Programming Research Group, Applied Science Division, International Business Machines Corporation |location=New York, USA |date=1954-11-10 |pages=4, 6 |url=https://archive.computerhistory.org/resources/text/Fortran/102679231.05.01.acc.pdf |access-date=2022-07-04 |url-status=live |archive-url=https://web.archive.org/web/20220329234459/https://archive.computerhistory.org/resources/text/Fortran/102679231.05.01.acc.pdf |archive-date=2022-03-29}} (29 pages)</ref>
<ref name="Sayre_1956">{{cite book |title=The FORTRAN Automatic Coding System for the IBM 704 EDPM: Programmer's Reference Manual |publisher=Applied Science Division and Programming Research Department, International Business Machines Corporation |location=New York, USA |date=1956-10-15 |editor-last=Sayre |editor-first=David |author-last1=Backus |author-first1=John Warner |author-last2=Beeber |author-first2=R. J. |author-last3=Best |author-first3=Sheldon F. |author-last4=Goldberg |author-first4=Richard |author-link4=Richard Goldberg |author-last5=Herrick |author-first5=Harlan L. |author-last6=Hughes |author-first6=R. A. |author-last7=Mitchell |author-first7=L. B. |author-last8=Nelson |author-first8=Robert A. |author-last9=Nutt |author-first9=Roy |author-link9=Roy Nutt |author-first10=David |author-last10=Sayre |author-link10=David Sayre |author-first11=Peter B. |author-last11=Sheridan |author-first12=Harold |author-last12=Stern |author-first13=Irving |author-last13=Ziller |page=15 |url=https://archive.computerhistory.org/resources/text/Fortran/102649787.05.01.acc.pdf |access-date=2022-07-04 |url-status=live |archive-url=https://web.archive.org/web/20220704193549/http://archive.computerhistory.org/resources/text/Fortran/102649787.05.01.acc.pdf |archive-date=2022-07-04}} (2+51+1 pages)</ref>
<ref name="Bronstein_1987">{{cite book |title=Taschenbuch der Mathematik |language=de |trans-title=Pocketbook of mathematics |title-link=Bronstein and Semendjajew |chapter=2.4.1.1. Definition arithmetischer Ausdrücke |trans-chapter=Definition of arithmetic expressions |author-last1=Bronstein |author-first1=Ilja Nikolaevič<!-- Nikolajewitsch --> |author-last2=Semendjajew |author-first2=Konstantin Adolfovič<!-- Adolfowitsch --> |author-link2=Konstantin Adolfovic Semendyayev<!-- 1908–1988 --> |editor-first1=Günter |editor-last1=Grosche |editor-first2=Viktor |editor-last2=Ziegler<!-- 1922–1980--> |editor-first3=Dorothea |editor-last3=Ziegler |others=Weiß, Jürgen<!-- lector --> |translator-last=Ziegler |translator-first=Viktor |volume=1 |date=1987 |edition=23 |orig-year=1945 |publisher=[[Company:Verlag Harri Deutsch|Verlag Harri Deutsch]] (and B. G. Teubner Verlagsgesellschaft, Leipzig) |publication-place=Thun, Switzerland / Frankfurt am Main, Germany |location=Leipzig, Germany |isbn=3-87144-492-8 |pages=115–120, 802 <!-- |quote=Regel 7: Ist ''F''(''A'') Teilzeichenreihe eines arithmetischen Ausdrucks oder einer seiner Abkürzungen und ''F'' eine Funktionenkonstante und ''A'' eine Zahlenvariable oder Zahlenkonstante, so darf ''F{{thin space}}A'' dafür geschrieben werden. [Darüber hinaus ist noch die Abkürzung ''F''<sup>''n''</sup>(''A'') für (''F''(''A''))<sup>''n''</sup> üblich. Dabei kann ''F'' sowohl Funktionenkonstante als auch Funktionenvariable sein.] -->}}</ref>
<ref name="NIST_2010">{{cite book |title=NIST Handbook of Mathematical Functions |title-link=NIST Handbook of Mathematical Functions |editor-last=Olver |editor-first=Frank W. J. |editor2-last=Lozier |editor2-first=Daniel W. |editor3-last=Boisvert |editor3-first=Ronald F. |editor4-last=Clark |editor4-first=Charles W. |date=2010 |publisher=[[National Institute of Standards and Technology]] (NIST), U.S. Department of Commerce, Cambridge University Press |isbn=978-0-521-19225-5 |mr=2723248}}[http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521140638]</ref>
<ref name="Zeidler_2013">{{cite book |title=Springer-Handbuch der Mathematik I |title-link=Springer-Handbuch der Mathematik |volume=I |language=de |editor-last=Zeidler |editor-first=Eberhard |author-last1=Zeidler |author-first1=Eberhard |author-last2=Schwarz |author-first2=Hans Rudolf |author-last3=Hackbusch |author-first3=Wolfgang |author-link3=Wolfgang Hackbusch |author-last4=Luderer |author-first4=Bernd |author-link4=:de:Bernd Luderer |author-last5=Blath |author-first5=Jochen |author-last6=Schied |author-first6=Alexander |author-last7=Dempe |author-first7=Stephan |author-last8=Wanka |author-first8=Gert |author-link8=Gert Wanka |author-last9=Hromkovič |author-first9=Juraj |author-link9=Juraj Hromkovič |author-last10=Gottwald |author-first10=Siegfried |author-link10=Siegfried Gottwald |publisher=Springer Spektrum, Springer Fachmedien Wiesbaden |location=Berlin / Heidelberg, Germany |edition=1 |date=2013 |orig-year=2012 |isbn=978-3-658-00284-8 |page=590<!-- |doi=10.1007/978-3-658-00285-5 |url=https://link.springer.com/book/10.1007/978-3-658-00285-5 |access-date=2024-01-11 -->}} (xii+635 pages)</ref>
<!-- <ref name="Stibitz_1957">{{cite book |title=Mathematics and Computers |author-last1=Stibitz |author-first1=George Robert |author-last2=Larrivee |author-first2=Jules A. |date=1957 |edition=1 |publisher=McGraw-Hill Book Company, Inc. |publication-place=New York, USA / Toronto, Canada / London, UK |location=Underhill, Vermont, USA |lccn=56-10331 |page=169}} (10+228 pages) (NB. Stibitz uses parentheses even in conjunction with trigonometric functions (like <code>(cos&nbsp;''u'')<sup>''n''</sup></code>) to avoid the ambiguity of the <code>cos<sup>''n''</sup>&nbsp;''u''</code> notation.)</ref> -->
<ref name="Cajori_1929">{{cite book |author-last=Cajori |author-first=Florian |title=A History of Mathematical Notations |volume=2 |orig-year=March 1929 |publisher=Open court publishing company |location=Chicago, USA |date=1952 |edition=3rd |pages=108, 176–179, 336, 346 |isbn=978-1-60206-714-1 |url=https://books.google.com/books?id=bT5suOONXlgC |access-date=2016-01-18}}</ref>
<!-- <ref name="Peirce_1852">{{cite book |author-last=Peirce |author-first=Benjamin |title=Curves, Functions and Forces |volume=I |edition=new |location=Boston, USA |date=1852 |page=203}}</ref>-->
<ref name="Peano_1903">{{cite book |author-last=Peano |author-first=Giuseppe |title=Formulaire mathématique |language=fr |volume=IV |date=1903 |page=229}}</ref>
<!-- <ref name="Pringsheim-Molk_1907">{{cite book |author-last1=Pringsheim |author-first1=Alfred |author-last2=Molk |author-first2=Jules |author-link2=Jules Molk |title=Encyclopédie des sciences mathématiques pures et appliquées |language=fr |id=Part I |volume=I |date=1907 |page=195}}</ref> -->
<ref name="Herschel_1813">{{cite journal |author-last=Herschel |author-first=John Frederick William |title=On a Remarkable Application of Cotes's Theorem |journal=Philosophical Transactions of the Royal Society of London |publisher=Royal Society of London, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall |location=London |volume=103 |number=Part 1 |date=1813 |orig-year=1812-11-12 |jstor=107384 |pages=8–26 [10] |s2cid=118124706 |doi=10.1098/rstl.1813.0005}}</ref>
<ref name="Herschel_1820">{{cite book |author-last=Herschel |author-first=John Frederick William |title=A Collection of Examples of the Applications of the Calculus of Finite Differences |chapter=Part III. Section I. Examples of the Direct Method of Differences |location=Cambridge, UK |publisher=Printed by J. Smith, sold by J. Deighton & sons |date=1820 |pages=1–13 [5–6] |chapter-url=https://books.google.com/books?id=PWcSAAAAIAAJ&pg=PA5 |access-date=2020-08-04 |url-status=live |archive-url=https://web.archive.org/web/20200804031020/https://books.google.de/books?id=PWcSAAAAIAAJ&jtp=5 |archive-date=2020-08-04}} [https://archive.org/details/acollectionexam00lacrgoog] (NB. Inhere, Herschel refers to his {{citeref|Herschel|1813|1813 work|style=plain}} and mentions Hans Heinrich Bürmann's older work.)</ref>
<ref name="Euler_1748">{{cite book |author-last=Euler |author-first=Leonhard |title=Introductio in analysin infinitorum |language=la |location=Lausanne |publisher=Marc-Michel Bousquet |date=1748 |volume=I |pages=69, 98–99 <!-- |quote-page=69 --> |url=https://gallica.bnf.fr/ark:/12148/bpt6k33510/f93.image |quote=Primum ergo considerandæ sunt quantitates exponentiales, seu Potestates, quarum Exponens ipse est quantitas variabilis. Perspicuum enim est hujusmodi quantitates ad Functiones algebraicas referri non posse, cum in his Exponentes non nisi constantes locum habeant.}}</ref>
<!-- <ref name="Euler_1760">{{cite journal |author-last=Euler |author-first=Leonhard |title=Novi commentarii Academiae Scientiarum Imperialis Petropolitanae, for the years 1754–1755 |language=la |location=Petropoli |date=1760 |orig-year=1754 |page=172}}</ref> -->
<!-- <ref name="Karsten_1760">{{cite book |author-last=Karsten |author-first=Wenceslaus Johann Gustav |title=Mathesis theoretica Elementaris Atque Sublimior |language=la |chapter=Sectio XIII. De sectionibus angulorum et arcuum circularium |location=Rostock |date=1760 |page=511 |url=https://www.digitale-sammlungen.de/de/view/bsb10594020?page=539 |access-date=2020-08-04}} [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10594020-6 ]</ref> -->
<!-- <ref name="Scherffer_1772">{{cite book |author-last=Scherffer |author-first=Karl "Carolo" |title=Institutionum analyticarum, pars secunda |language=la |location=Vienna |date=1772 |page=144}}</ref> -->
<!-- <ref name="Frisius_1782">{{cite book |author-last=Frisius (Frisii) |author-first=Paulli |title=Operum tomus primus |language=la |location=Milano |date=1782 |page=303}}</ref> -->
<!-- <ref name="Abel_1826">{{cite journal |author-last=Abel |author-first=Niels Henrik |language=de |journal=Journal für die reine und angewandte Mathematik |publisher=August Leopold Crelle |volume=I |location=Berlin |date=1826 |pages=318–337 |postscript=;}} {{cite journal |author-last=Abel |author-first=Niels Henrik |author-link=Niels Henrik Abel |language=de |journal=Journal für die reine und angewandte Mathematik |publisher=August Leopold Crelle |volume=II |location=Berlin |date=1827 |page=26}}</ref> -->
<!-- <ref name="Ohm_1829">{{cite book |author-last=Ohm |author-first=Martin |title=System der Mathematik |language=de |id=Part 3 |location=Berlin |date=1829 |page=21}}</ref> -->
<!-- <ref name="Cagnoli_1786">{{cite book |author-last=Cagnoli |author-first=Antonio |title=Traité de Trigonométrie |language=fr |publisher=trad. par Chompré |location=Paris |date=1786 |page=20}}</ref> -->
<!-- <ref name="DeMorgan_1849">{{cite book |author-last=De Morgan |author-first=Augustus |title=Trigonometry and Double Algebra |location=London |date=1849 |page=35}}</ref> -->
<!-- <ref name="Serret_1857">{{cite book |author-last=Serret |author-first=Joseph Alfred |title=Traité de Trigonométrie |language=fr |edition=2nd |location=Paris |date=1857 |page=12}}</ref> -->
<!-- <ref name="Todhunter_1876">{{cite book |author-last=Todhunter |author-first=Isaac |title=Plane Trigonometry |edition=6th |location=London |date=1876 |page=19}}</ref> -->
<!-- <ref name="Hobson_1911">{{cite book |author-last=Hobson |author-first=Ernest William |title=Treatise on Plane Trigonometry |location=Cambridge, UK |date=1911 |page=19}}</ref> -->
<!-- <ref name="Toledo_1917">{{cite book |author-last=de Toledo |author-first=Luis Octavio |title=Tradado de Trigonometria |language=es |edition=3rd |location=Madrid |date=1917 |page=64}}</ref> -->
<!-- <ref name="Rothe_1921">{{cite book |author-last=Rothe |author-first=Hermann |title=Vorlesungen über höhere Mathematik |language=de |location=Vienna |date=1921 |page=261}}</ref> -->
}}

{{Hyperoperations}}
{{Orders of magnitude (time)}}
{{Classes of natural numbers}}

[[Category:Exponentials]]
[[Category:Binary operations]]
[[Category:Unary operations]]

{{Sourceattribution|Exponentiation}}