Editing Exponentiation (section)

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

The exponentiation operation with integer exponents may be defined directly from elementary [[arithmetic operation]]s.

===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]] convention, which may be used in every [[algebraic structure]] with a multiplication that has an [[multiplicative identity|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}}:
: <math>b^{-n} = \frac{1}{b^n}</math>.<ref name=":1"/>
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (<math>\infty</math>).<ref>{{cite book
 | last = Knobloch | first = Eberhard | author-link = Eberhard Knobloch
 | 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]]s in a multiplicative [[monoid]], that is, an [[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]] {{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]] 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]], many [[algorithm]]s involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose [[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===
{{See also|#Exponentiation over sets|l1=Exponentiation over sets}}

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]]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]]) 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]] to denote large or small numbers. For instance, {{val|299792458|u=m/s}} (the [[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]], since a set with {{math|''n''}} members has a [[power set]], the set of all of its [[subset]]s, which has {{math|2<sup>''n''</sup>}} members.

Integer powers of {{math|2}} are important in [[computer science]]. The positive integer powers {{math|2<sup>''n''</sup>}} give the number of possible values for an {{math|''n''}}-[[bit]] integer [[binary number]]; for example, a [[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]]), 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]]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]] 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]] 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]], 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]] 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 (mathematics)|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]] {{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 (mathematics)|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 function]]s.

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

===Table of powers of decimal digits===
{|class="wikitable" style="text-align:right"
! ''n'' !! ''n''<sup>2</sup> !! ''n''<sup>3</sup> !! ''n''<sup>4</sup> !! ''n''<sup>5</sup> !! ''n''<sup>6</sup> !! ''n''<sup>7</sup> !! ''n''<sup>8</sup> !! ''n''<sup>9</sup> !! ''n''<sup>10</sup>
|-
|'''1''' || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
|'''2''' || 4 || 8 || 16 || 32 || 64 || 128 || 256 || 512 || 1024
|-
|'''3''' || 9 || 27 || 81 || 243 || 729 || {{val|2,187|fmt=gaps}} || {{val|6,561|fmt=gaps}} || {{val|19,683}} || {{val|59,049}}
|-
|'''4''' || 16 || 64 || 256 || {{val|1024|fmt=gaps}} || {{val|4,096|fmt=gaps}} || {{val|16,384}} || {{val|65,536}} || {{val|262,144}} || {{val|1,048,576}}
|-
|'''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}}
|-
|'''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}}
|-
|'''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}}
|-
|'''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}}
|-
|'''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}}
|-
|'''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}}
|}