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| | The Wildcats' rebuilding program has hit a snag due to be able to lack of offensive firepower. Now they draw just one of the West's toughest out-of-conference mid-major-type programs their Cougars and desperately need to have a win or Arizona fans will be manufactured even more restless before Pac-10 play the game of.<br><br> |
| '''Exponentiation''' is a [[mathematics|mathematical]] [[operation (mathematics)|operation]], written as '''''b''<sup>''n''</sup>''', involving two numbers, the '''[[Base (exponentiation)|base]]''' ''b'' and the '''exponent''' (or '''index''' or '''power''') ''n''. When ''n'' is a [[positive integer]], exponentiation corresponds to repeated [[multiplication]]; in other words, a product of '''''n''''' factors, each of which is equal to '''''b''''' (the product itself can also be called '''power'''): | |
| :<math>b^n = \underbrace{b \times \cdots \times b}_n</math>
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| just as multiplication by a positive integer corresponds to repeated [[addition]]:
| | <br><br>Be a Pollyanna---In other words, surf the glass half full, instead of half write off. As some patients may be undergoing depression, it's important for in order to encourage associated with.<br><br>A stone wall for your household landscape may also serve an extraordinarily practical concept. It can be used to discover your property from the neighbouring units. It adds some semblance of privacy whether or not it is only a low outlet. It looks naturally beautiful in contrast with building a concrete wall to divide your home and property. To enhance the beauty of the wall, you're able to plant some vines around it and let it creep near the wall. Flowers and hedges can additionally be arranged beautifully along the wall.<br><br>Lavender leaves are also strongly fragrant and may even be sticky with essential oils. The flowers are great in fresh bouquets. These people also be used for flavoring in salads or vinegars. Dried bouquets and flowers are used in crafts and as home work environment.<br><br>This could be very helpful. When talking about possible solutions to the war against terrorism, Clinton said "most of major things existence are simple".<br><br>Clinton used the metaphor of the gap between the invention of the club and the shield to describe the present situation within the war against terrorism. He explained "this gap needs to closed". Metaphors can give intangible concepts more impact with a large group.<br><br>When purchasing a home, by using your every monthly payment, tend to be putting money into deals. On the contrary, when you are renting a home you have to give money to the owner it is actually definitely an expense. Any time you pay a mortgage, a percentage goes toward your worth. Indeed, this is similar to having money inside your bank account, which you'll be able to draw upon later when necessary. Every year, rental rates increase as well as the principle on mortgage is decreasing with every payment. Further, the housing marketplace continues to cultivate and hence the valuations of housing.<br><br>Personalized baby blankets additionally trendy here. This will enable your baby to exercise his individualism and creativity. Now, your baby will generally be in a snugly mood with a warm blanket wrapping him around during cold winter nights.<br><br>If you adored this post and you would like to receive more details relating to [http://www.hopesgrovenurseries.co.uk/ www.hopesgrovenurseries.co.uk] kindly see our web page. |
| :<math>b \times n = \underbrace{b + \cdots + b}_n</math>
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| The exponent is usually shown as a [[superscript]] to the right of the base. The exponentiation ''b''<sup>''n''</sup> can be read as: '''''b''' raised to the '''n'''-th power'', '''''b''' raised to the power of '''n''''', or '''''b''' raised by the exponent of '''n''''', most briefly as '''''b''' to the '''n'''''. Some exponents have their own pronunciation: for example, ''b''<sup>2</sup> is usually read as '''''b''' squared'' and ''b''<sup>3</sup> as '''''b''' cubed''.
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| The power ''b''<sup>''n''</sup> can be defined also when ''n'' is a negative integer, for nonzero ''b''. No natural extension to all real ''b'' and ''n'' exists, but when the base ''b'' is a positive real number, ''b''<sup>''n''</sup> can be defined for all real and even complex exponents ''n'' via the [[exponential function]] ''e''<sup>''z''</sup>. [[trigonometry|Trigonometric functions]] can be expressed in terms of complex exponentiation.
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| Exponentiation where the [[Matrix exponential|exponent is a matrix]] is used for solving systems of [[linear differential equation]]s.
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| Exponentiation is used pervasively in many other fields, including economics, biology, chemistry, physics, as well as computer science, with applications such as [[compound interest]], [[population growth]], chemical [[reaction kinetics]], [[wave]] behavior, and [[public key cryptography]].
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| [[Image:Expo02.svg|thumb|315px|Graphs of {{nowrap|1=''y'' = ''b''<sup>''x''</sup>}} for various bases ''b'': [[#Powers of ten|base 10]] (<span style="color:green">green</span>), [[#The exponential function|base ''e'']] (<span style="color:red">red</span>), [[#Powers of two|base 2]] (<span style="color:blue">blue</span>), and base {{sfrac|2}} (<span style="color:cyan">cyan</span>). Each curve passes through the point {{nowrap|(0, 1)}} because any nonzero number raised to the power of 0 is 1. At {{nowrap|1=''x'' = 1}}, the value of ''y'' equals the base because any number raised to the power of 1 is the number itself.]]
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| {{Calculation results}}
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| ==Background and terminology==
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| The expression ''b''<sup>2</sup> = ''b''·''b'' is called the [[Square (algebra)|square]] of ''b'' because the area of a square with side-length ''b'' is ''b''<sup>2</sup>. It is pronounced "b squared".
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| The expression ''b''<sup>3</sup> = ''b''·''b''·''b'' is called the [[Cube (algebra)|cube]] of ''b'' because the volume of a cube with side-length ''b'' is ''b''<sup>3</sup>. It is pronounced "b cubed".
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| The exponent says how many copies of the base are multiplied together. For example, 3<sup>5</sup> = 3·3·3·3·3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the ''base'', 5 is the ''exponent'', and 243 is the ''power'' or, more specifically, ''the fifth power of 3'', ''3 raised to the fifth power'', or ''3 to the power of 5''.
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| The word "raised" is usually omitted, and very often "power" as well, so 3<sup>5</sup> is typically pronounced "three to the fifth" or "three to the five".
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| Exponentiation may be generalized from integer exponents to more general types of numbers.
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| ==Integer exponents==
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| The exponentiation operation with integer exponents requires only [[elementary algebra]].
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| ===Positive integer exponents===
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| Formally, powers with positive integer exponents may be defined by the initial condition<ref>{{cite book |url=http://books.google.com/books?id=qToTAgAAQBAJ&lpg=PA93&pg=PA94#v=onepage&f=false |title=Abstract Algebra: an inquiry based approach |first1=Jonathan K. |last1=Hodge |first2=Steven |last2=Schlicker |first3=Ted |last3=Sundstorm |page=94 |year=2014 |publisher=CRC Press |isbn=978-1-4665-6706-1}}</ref>
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| :<math>b^1 = b</math>
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| and the [[recurrence relation]]
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| :<math>b^{n+1} = b^n \cdot b</math>
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| From the [[Associative property|associativity]] of multiplication, it follows that for any positive integers ''m'' and ''n'',
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| :<math>b^{m+n} = b^m \cdot b^n</math>
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| ===Zero exponent===
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| Any nonzero number raised by the exponent 0 is 1;<ref>{{cite book|url=http://books.google.com/books?id=YOdtemSmzQQC&pg=PA101&hl=en&sa=X&ei=HIbQUqbuOYSO7QbXpIGIBw&ved=0CDkQ6AEwATgo#v=onepage&f=false |title=Technical Shop Mathematics |first1=Thomas |last1=Achatz |page=101 |year=2005 |edition=3rd |publisher=Industrial Press |isbn=0-8311-3086-5}}</ref> one interpretation of such a power is as an [[empty product]]. The case of 0<sup>0</sup> is discussed [[#Zero to the power of zero|below]].
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| ===Negative exponents===
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| The following identity holds for an arbitrary integer ''n'' and nonzero ''b'':
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| :<math>b^{-n} = 1/b^n </math>
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| Raising 0 by a negative exponent is left undefined.
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| The identity above may be derived through a definition aimed at extending the range of exponents to negative integers.
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| For non-zero ''b'' and positive ''n'', the recurrence relation from the previous subsection can be rewritten as
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| :<math>b^{n} = {b^{n+1}}/{b}, \quad n \ge 1 .</math>
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| By defining this relation as valid for all integer ''n'' and nonzero ''b'', it follows that
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| :<math>\begin{align}
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| b^0 &= {b^{1}}/{b} = 1 \\
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| b^{-1} &= {b^{0}}/{b} = {1}/{b}
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| \end{align}</math>
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| and more generally for any nonzero ''b'' and any nonnegative integer ''n'',
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| :<math>b^{-n} = {1}/{b^n} .</math>
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| This is then readily shown to be true for every integer ''n''.
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| ===Combinatorial interpretation===
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| For nonnegative integers ''n'' and ''m'', the power ''n''<sup>''m''</sup> equals the [[cardinality]] of the set of ''m''-[[tuple]]s from an ''n''-element [[Set (mathematics)|set]], or the number of ''m''-letter words from an ''n''-letter alphabet.
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| :{|
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| |-
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| | 0<sup>5</sup> = │ {} │ = 0
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| | There is no [[5-tuple]] from the empty set.
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| |-
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| | 1<sup>4</sup> = │ { (1,1,1,1) } │ = 1
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| | There is one [[4-tuple]] from a one-element set.
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| |-
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| | 2<sup>3</sup> = │ { (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) } │ = 8
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| | There are eight [[3-tuple]]s from a two-element set.
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| |-
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| | 3<sup>2</sup> = │ { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } │ = 9
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| | There are nine [[2-tuple]]s from a three-element set.
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| |-
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| | 4<sup>1</sup> = │ { (1), (2), (3), (4) } │ = 4
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| | There are four [[1-tuple]]s from a four-element set.
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| |-
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| | 5<sup>0</sup> = │ { () } │ = 1
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| | There is exactly one [[0-tuple]].
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| |}
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| {{see also|#Exponentiation over sets|l1=Exponentiation over sets}}
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| ===Identities and properties===
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| The following [[identity (mathematics)|identities]] hold for all integer exponents, provided that the base is non-zero:
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| :<math>\begin{align}
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| b^{m + n} &= b^m \cdot b^n \\
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| (b^m)^n &= b^{m\cdot n} \\
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| (b \cdot c)^n &= b^n \cdot c^n
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| \end{align}</math>
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| Exponentiation is not [[commutative]]. This contrasts with addition and multiplication, which are. For example, {{nowrap|1=2 + 3 = 3 + 2 = 5}} and {{nowrap|1=2 · 3 = 3 · 2 = 6}}, but {{nowrap|1=2<sup>3</sup> = 8}}, whereas {{nowrap|1=3<sup>2</sup> = 9}}.
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| Exponentiation is not [[associative]] either. Addition and multiplication are. For example,
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| {{nowrap|1=(2 + 3) + 4 = 2 + (3 + 4) = 9}} and {{nowrap|1=(2 · 3) · 4 = 2 · (3 · 4) = 24}}, but 2<sup>3</sup> to the 4 is 8<sup>4</sup> or 4,096, whereas 2 to the 3<sup>4</sup> is 2<sup>81</sup> or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up:
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| :<math>b^{p^q} = b^{(p^q)} \ne (b^p)^q = b^{(p \cdot q)} = b^{p \cdot q} .</math>
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| ===Particular bases===
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| ====Powers of ten====
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| {{see also|Scientific notation}}
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| In the base ten ([[decimal]]) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, {{val|e=3}} = 1,000 and {{val|e=-4}} = 0.0001.
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| Exponentiation with base [[10 (number)|10]] is used in [[scientific notation]] to denote large or small numbers. For instance, 299,792,458 m/s (the [[speed of light]] in vacuum, in [[metre 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}}.
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| [[SI prefix]]es based on powers of 10 are also used to describe small or large quantities. For example, the prefix [[Kilo-|kilo]] means {{nowrap|1={{val|e=3}} = 1,000}}, so a kilometre is 1,000 metres.
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| ====Powers of two====
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| The positive [[Power of two|powers of 2]] are important in [[computer science]] because there are 2<sup>''n''</sup> possible values for an ''n''-[[bit]] [[binary numeral system|binary]] [[variable (programming)|variable]].
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| Powers of 2 are important in [[set theory]] since a set with ''n'' members has a [[power set]], or set of all [[subset]]s of the original set, with 2<sup>''n''</sup> members.
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| The negative powers of 2 are commonly used, and the first two have special names: ''[[One half|half]]'', and ''[[4 (number)|quarter]]''.
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| In the base 2 (binary) number system, integer powers of 2 are written as 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, two to the power of three is written as 1000 in binary.
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| ====Powers of one====
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| The integer powers of one are all one: {{nowrap|1=1<sup>''n''</sup> = 1}}.
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| ====Powers of zero====
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| If the exponent is positive, the power of zero is zero: {{nowrap|1=0<sup>''n''</sup> = 0}}, where {{nowrap|1=''n'' > 0}}.
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| If the exponent is negative, the power of zero (0<sup>''n''</sup>, where ''n'' < 0) is undefined, because division by zero is implied.
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| If the exponent is zero, some authors define {{nowrap|0<sup>0</sup> {{=}} 1}}, whereas others leave it undefined, as discussed [[#Zero to the power of zero|below]].
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| ====Powers of minus one====
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| If ''n'' is an even integer, then (−1)<sup>''n''</sup> = 1.
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| If ''n'' is an odd integer, then (−1)<sup>''n''</sup> = −1.
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| Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number ''i'', see the section on [[#Powers of complex numbers|Powers of complex numbers]].
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| ===Large exponents===
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| The [[limit of a sequence]] of powers of a number greater than one diverges, in other words they grow without bound:
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| :''b''<sup>''n''</sup> → ∞ as ''n'' → ∞ when ''b'' > 1
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| 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".
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| Powers of a number with [[absolute value]] less than one tend to zero:
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| :''b''<sup>''n''</sup> → 0 as ''n'' → ∞ when |''b''| < 1
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| Any power of one is always itself:
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| :''b''<sup>''n''</sup> = 1 for all ''n'' if ''b'' = 1
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| If the number ''b'' varies tending to 1 as the exponent tends to infinity then the limit is not necessarily one of those above. A particularly important case is
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| :(1 + 1/''n'')<sup>''n''</sup> → ''e'' as ''n'' → ∞
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| See the section below, [[#The exponential function|The exponential function]].
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| Other limits, in particular of those tending to [[indeterminate forms]], are described in [[#Limits of powers|limits of powers]] below.
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| ==Rational exponents==
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| {{Main|nth root}}
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| [[Image:Root graphs.svg|right|thumb|300px|From top to bottom: ''x''<sup>1/8</sup>, ''x''<sup>1/4</sup>, ''x''<sup>1/2</sup>, ''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>4</sup>, ''x''<sup>8</sup>.]]
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| An ''' ''n''-th root''' of a [[number]] ''b'' is a number ''x'' such that ''x''<sup>''n''</sup> = ''b''.
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| If ''b'' is a positive real number and ''n'' is a positive integer, then there is exactly one positive real solution to ''x<sup>n</sup>'' = ''b''. This solution is called the '''principal [[Nth root|''n''-th root]]''' of ''b''. It is denoted <sup>''n''</sup>√<span style="text-decoration:overline">''b''</span>, where √<span style="text-decoration:overline"> </span> is the '''radical''' [[symbol]]; alternatively, it may be written ''b''<sup>1/''n''</sup>. For example: 4<sup>1/2</sup> = 2, 8<sup>1/3</sup> = 2.
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| This follows from noting that
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| :<math>x^n = \underbrace{ b^\frac{1}{n} \times b^\frac{1}{n} \times \cdots \times b^\frac{1}{n} }_n = b^{\left( \frac{1}{n} + \frac{1}{n} + \cdots + \frac{1}{n} \right)} = b^\frac{n}{n} = b^1 = b</math>
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| If ''n'' is [[Parity (mathematics)|even]], then ''x<sup>n</sup>'' = ''b'' has two real solutions if ''b'' is positive, which are the positive and negative ''n''th roots. The equation has no solution in real numbers if ''b'' is negative.
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| If ''n'' is odd, then ''x<sup>n</sup>'' = ''b'' has one real solution. The solution is positive if ''b'' is positive and negative if ''b'' is negative.
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| [[Rational number|Rational]] powers ''m''/''n'', where ''m''/''n'' is in [[lowest terms]], are positive if ''m'' is even, negative for negative ''b'' if ''m'' and ''n'' are odd, and can be either sign if ''b'' is positive and ''n'' is even. (−27)<sup>1/3</sup> = −3, (−27)<sup>2/3</sup> = 9, and 4<sup>3/2</sup> has two roots 8 and −8. Since there is no real number ''x'' such that ''x''<sup>2</sup> = −1, the definition of ''b''<sup>''m''/''n''</sup> when ''b'' is negative and ''n'' is even must use the [[imaginary unit]] ''i'', as described more fully in the section [[#Powers of complex numbers|Powers of complex numbers]].
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| A power of a positive real number ''b'' with a rational exponent ''m''/''n'' in lowest terms satisfies
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| :<math>b^\frac{m}{n} = \left(b^m\right)^\frac{1}{n} = \sqrt[n]{b^m}</math>
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| where ''m'' is an integer and ''n'' is a positive integer.
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| Care needs to be taken when applying the power law identities with negative ''n''th roots. For instance,
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| −27 = (−27)<sup>((2/3)⋅(3/2))</sup> = ((−27)<sup>2/3</sup>)<sup>3/2</sup> = 9<sup>3/2</sup> = 27 is clearly wrong. The problem here occurs in taking the positive square root rather than the negative one at the last step, but in general the same sorts of problems occur as described for complex numbers in the section [[#Failure of power and logarithm identities|Failure of power and logarithm identities]].
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| ==Real exponents==
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| The [[#Identities and properties|identities and properties]] shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity
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| :<math>(b^r)^s = b^{r\cdot s}</math>
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| cannot be extended consistently to where ''b'' is a negative real number, see [[#Real exponents with negative bases|Real exponents with negative bases]]. The failure of this identity is the basis for the problems with complex number powers detailed under [[#Failure of power and logarithm identities|failure of power and logarithm identities]].
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| The extension of exponentiation to real powers of positive real numbers can be done either by extending the rational powers to reals by continuity, or more usually as given in the section [[#Powers via logarithms|Powers via logarithms]] below.
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| ===Limits of rational exponents===
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| Since any [[irrational number]] can be approximated by a rational number, exponentiation of a positive real number ''b'' with an arbitrary real exponent ''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 |year=2011 |pages=278–283 |isbn=978-0-7637-7947-4}}</ref>
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| :<math> b^x = \lim_{r \to x} b^r\quad(r\in\mathbb Q,\,x\in\mathbb R)</math>
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| where the limit as ''r'' gets close to ''x'' is taken only over rational values of ''r''. This limit only exists for positive ''b''. The [[(ε, δ)-definition of limit]] is used, this involves showing that for any desired accuracy of the result <math>\scriptstyle b^x</math> one can choose a sufficiently small interval around {{mvar|x}} so all the rational powers in the interval are within the desired accuracy.
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| For example, if <math>\scriptstyle x \;=\; \pi</math>, the nonterminating decimal representation <math>\scriptstyle \pi \;=\; 3.14159\ldots</math> can be used (based on strict monotonicity of the rational power) to obtain the intervals bounded by rational powers
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| :<math>[b^3,b^4]</math>, <math>[b^{3.1},b^{3.2}]</math>, <math>[b^{3.14},b^{3.15}]</math>, <math>[b^{3.141},b^{3.142}]</math>, <math>[b^{3.1415},b^{3.1416}]</math>, <math>[b^{3.14159},b^{3.14160}]</math>, …
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| The bounded intervals converge to a unique real number, denoted by <math>\scriptstyle b^\pi</math>. This technique can be used to obtain any irrational power of {{mvar|b}}. The function <math>\scriptstyle f(x) \;=\; b^x</math> is thus defined for any real number {{mvar|x}}.
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| ===The exponential function===
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| {{Main|Exponential function}}
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| The important mathematical constant [[E (mathematical constant)|{{mvar|e}}]], sometimes called [[Euler's number]], is approximately equal to 2.718 and is the base of the [[natural logarithm]]. Although exponentiation of ''e'' could, in principle, be treated the same as exponentiation of any other real number, such exponentials turn out to have particularly elegant and useful properties. Among other things, these properties allow exponentials of ''e'' to be generalized in a natural way to other types of exponents, such as complex numbers or even matrices, while coinciding with the familiar meaning of exponentiation with rational exponents.
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| As a consequence, the notation ''e''<sup>''x''</sup> usually denotes a generalized exponentiation definition called the '''exponential function''', exp(''x''), which can be defined [[Characterizations of the exponential function|in many equivalent ways]], for example by:
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| :<math>\exp(x) = \lim_{n \rightarrow \infty} \left(1+\frac x n \right)^n </math>
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| Among other properties, exp satisfies the exponential identity:
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| :<math>\exp(x+y) = \exp(x) \cdot \exp(y)</math>
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| The exponential function is defined for all integer, fractional, real, and [[complex number|complex]] values of {{mvar|x}}. It can even be used to extend exponentiation to some nonnumerical entities such as [[matrix exponential|square matrices]] (in which case the exponential identity only holds when {{mvar|x}} and {{mvar|y}} commute).
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| Since <math>\scriptstyle \exp(1)</math> is equal to {{mvar|e}} and <math>\scriptstyle \exp(x)</math> satisfies the exponential identity, it immediately follows that exp(''x'') coincides with the repeated-multiplication definition of ''e''<sup>''x''</sup> for integer ''x'', and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the ''e''<sup>''x''</sup> definitions in the previous section for all real ''x'' by continuity.
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| ===Powers via logarithms===
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| The [[natural logarithm]] ln(''x'') is the [[inverse function|inverse]] of the exponential function ''e''<sup>''x''</sup>. It is defined for ''b'' > 0, and satisfies
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| :<math>b = e^{\ln b}</math>
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| If ''b''<sup>''x''</sup> is to preserve the logarithm and exponent rules, then one must have
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| :<math>b^x = (e^{\ln b})^x = e^{x \cdot\ln b}</math>
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| for each real number ''x''.
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| This can be used as an alternative definition of the real number power ''b''<sup>''x''</sup> and agrees with the definition given above using rational exponents and continuity. The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below.
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| ===Real exponents with negative bases===
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| Powers of a positive real number are always positive real numbers. The solution of x<sup>2</sup> = 4, however, can be either 2 or −2. The principal value of 4<sup>1/2</sup> is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well behaved.
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| Neither the logarithm method nor the rational exponent method can be used to define ''b''<sup>''r''</sup> as a real number for a negative real number ''b'' and an arbitrary real number ''r''. Indeed, ''e''<sup>''r''</sup> is positive for every real number ''r'', so ln(''b'') is not defined as a real number for ''b'' ≤ 0.
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| The rational exponent method cannot be used for negative values of ''b'' because it relies on [[continuous function|continuity]]. The function ''f''(''r'') = ''b''<sup>''r''</sup> has a unique continuous extension<ref name=Denlinger /> from the rational numbers to the real numbers for each ''b'' > 0. But when ''b'' < 0, the function ''f'' is not even continuous on the set of rational numbers ''r'' for which it is defined.
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| For example, consider ''b'' = −1. The ''n''th root of −1 is −1 for every odd natural number ''n''. So if ''n'' is an odd positive integer, (−1)<sup>(''m''/''n'')</sup> = −1 if ''m'' is odd, and (−1)<sup>(''m''/''n'')</sup> = 1 if ''m'' is even. Thus the set of rational numbers ''q'' for which (−1)<sup>''q''</sup> = 1 is [[dense set|dense]] in the rational numbers, as is the set of ''q'' for which (−1)<sup>''q''</sup> = −1. This means that the function (−1)<sup>''q''</sup> is not continuous at any rational number ''q'' where it is defined.
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| On the other hand, arbitrary [[#Powers of complex numbers|complex powers]] of negative numbers ''b'' can be defined by choosing a [[complex logarithm|''complex'' logarithm]] of ''b''.
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| ==Complex exponents with positive real bases==
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| ===Imaginary exponents with base e===
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| {{Main|Exponential function}}
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| [[Image:ExpIPi.gif|300px|thumb|right|The [[exponential function]] ''e''<sup>''z''</sup> can be defined as the [[limit of a sequence|limit]] of {{nowrap|(1 + ''z''/''N'')<sup>''N''</sup>}}, as ''N'' approaches infinity, and thus ''e''<sup>''iπ''</sup> is the limit of {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}}. In this animation ''N'' takes various increasing values from 1 to 100. The computation of {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}} is displayed as the combined effect of ''N'' repeated multiplications in the [[complex plane]], with the final point being the actual value of {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}}. It can be seen that as ''N'' gets larger {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}} approaches a limit of −1. Therefore, {{nowrap|1=''e''<sup>''iπ''</sup> = −1,}} which is known as [[Euler's identity]].]]
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| The geometric interpretation of the operations on [[complex numbers]] and the definition of the exponential function is the clue to understanding ''e''<sup>''ix''</sup> for real ''x''. Consider the [[right triangle]] {{nowrap|(0, 1, 1 + ''ix''/''n'').}} For big values of ''n'' the triangle is almost a [[circular sector]] with a small central angle equal to ''x''/''n'' [[radian]]s. The triangles {{nowrap|(0, (1 + ''ix''/''n'')<sup>''k''</sup>, (1 + ''ix''/''n'')<sup>''k''+1</sup>)}} are mutually [[Similar triangles|similar]] for all values of ''k''. So for large values of ''n'' the limiting point of {{nowrap|(1 + ''ix''/''n'')<sup>''n''</sup>}} is the point on the [[unit circle]] whose angle from the positive real axis is ''x'' radians. The [[polar coordinates]] of this point are {{nowrap|1=(''r'', ''θ'') = (1, ''x''),}} and the [[cartesian coordinate]]s are (cos ''x'', sin ''x''). So {{nowrap|1=''e''<sup> ''ix''</sup> = cos ''x'' + ''i''sin ''x'',}} and this is [[Euler's formula]], connecting [[algebra]] to [[trigonometry]] by means of [[complex number]]s.
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| The solutions to the equation ''e''<sup>''z''</sup> = 1 are the integer multiples of 2π''i'':
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| :<math>\{ z : e^z = 1 \} = \{ 2k\pi i : k \in \mathbb{Z} \}</math>
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| More generally, if e<sup>''v''</sup> = ''w'', then every solution to ''e''<sup>''z''</sup> = ''w'' can be obtained by adding an integer multiple of 2π''i'' to ''v'':
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| :<math>\{ z : e^z = w \} = \{ v + 2k\pi i : k \in \mathbb{Z} \}</math>
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| Thus the complex exponential function is a [[periodic function]] with period 2π''i''.
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| More simply: ''e''<sup>''iπ''</sup> = −1; ''e''<sup>''x'' + ''iy''</sup> = ''e''<sup>''x''</sup>(cos ''y'' + ''i'' sin ''y'').
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| ===Trigonometric functions===
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| {{Main|Euler's formula}}
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| It follows from Euler's formula stated above that the [[trigonometric functions]] cosine and sine are
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| :<math>\cos(z) = \frac{e^{iz} + e^{-iz}}{2}; \qquad \sin(z) = \frac{e^{iz} - e^{-iz}}{2i}</math>
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| Historically, cosine and sine were defined geometrically before the invention of complex numbers. The above formula reduces the complicated formulas for [[Trigonometric addition formulas|trigonometric functions of a sum]] into the simple exponentiation formula
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| :<math>e^{i(x+y)}=e^{ix}\cdot e^{iy}</math>
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| Using exponentiation with complex exponents may reduce problems in trigonometry to algebra.
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| ===Complex exponents with base e===
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| The power {{nowrap|''z'' {{=}} ''e''<sup>''x'' + ''iy''</sup>}} can be computed as ''e''<sup>''x''</sup> · ''e''<sup>''iy''</sup>. The real factor ''e''<sup>''x''</sup> is the [[absolute value]] of ''z'' and the complex factor ''e''<sup>''iy''</sup> identifies the [[direction (geometry, geography)|direction]] of ''z''.
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| ===Complex exponents with positive real bases===
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| If ''b'' is a positive real number, and ''z'' is any complex number, the power ''b''<sup>''z''</sup> is defined as ''e''<sup>''z''·ln(''b'')</sup>, where ''x'' = ln(''b'') is the unique real solution to the equation ''e''<sup>''x''</sup> = ''b''. So the same method working for real exponents also works for complex exponents.
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| For example:
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| :2<sup>''i''</sup> = ''e''<sup> ''i''·ln(2)</sup> = cos(ln(2)) + ''i''·sin(ln(2)) ≈ 0.76924 + 0.63896''i''
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| :''e''<sup>''i''</sup> ≈ 0.54030 + 0.84147''i''
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| :10<sup>''i''</sup> ≈ −0.66820 + 0.74398''i''
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| :(''e''<sup>2π</sup>)<sup>''i''</sup> ≈ 535.49<sup>''i''</sup> ≈ 1
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| The identity <math>(b^z)^u=b^{zu}</math> is not generally valid for complex powers. A simple counterexample is given by:
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| :<math>(e^{2\pi i})^i=1^i=1\neq e^{-2\pi}=e^{2\pi i\cdot i}</math>
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| The identity is, however, valid when <math>z</math> is a real number, and also when <math>u</math> is an integer.
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| ==Powers of complex numbers==
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| Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If ''i'' is the [[imaginary unit]] and ''n'' is an integer, then ''i''<sup>''n''</sup> equals 1, ''i'', −1, or −''i'', according to whether the integer ''n'' is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of ''i'' are useful for expressing [[sequence]]s of [[Root of unity#Periodicity|period 4]].
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| Complex powers of positive reals are defined via ''e''<sup>''x''</sup> as in section [[#Complex powers of positive real numbers|Complex powers of positive real numbers]] above. These are continuous functions.
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| Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or [[multivalued function]]s. Neither of these options is entirely satisfactory.
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| The rational power of a complex number must be the solution to an algebraic equation. Therefore it always has a finite number of possible values. For example, ''w'' = ''z''<sup>1/2</sup> must be a solution to the equation ''w''<sup>2</sup> = ''z''. But if ''w'' is a solution, then so is −''w'', because (−1)<sup>2</sup> = 1. A unique but somewhat arbitrary solution called the [[principal value]] can be chosen using a general rule which also applies for nonrational powers.
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| Complex powers and logarithms are more naturally handled as single valued functions on a [[Riemann surface]]. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a [[branch cut]]. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray.
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| Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the [[complex logarithm]] (see [[#Complex logarithms|below]]). The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as for the corresponding real numbers.
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| Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However in the common case of a positive real number the principal value is the same.
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| The powers of negative real numbers are not always defined and are discontinuous even where defined. In fact, they are only defined when the exponent is a rational number with the denominator being an odd integer. When dealing with complex numbers the complex number operation is normally used instead.
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| ===Complex exponents with complex bases===
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| For complex numbers ''w'' and ''z'' with ''w'' ≠ 0, the notation ''w''<sup>''z''</sup> is ambiguous in the same sense that [[complex logarithm|log ''w'']] is.
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| To obtain a value of ''w''<sup>''z''</sup>, first choose a logarithm of ''w''; call it log ''w''. Such a choice may be the [[complex logarithm#Definition of principal value|principal value]] Log ''w'' (the default, if no other specification is given), or perhaps a value given by some other [[complex logarithm#Branches of the complex logarithm|branch of log ''w'']] fixed in advance. Then, using the complex exponential function one defines
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| :<math>w^z = e^{z \log w}</math>
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| because this agrees with the [[#Real exponents|earlier definition]] in the case where ''w'' is a positive real number and the (real) principal value of log ''w'' is used.
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| If ''z'' is an [[integer]], then the value of ''w''<sup>''z''</sup> is independent of the choice of log ''w'', and it agrees with the [[#Positive integer exponents|earlier definition of exponentation with an integer exponent]].
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| If ''z'' is a [[rational number]] ''m''/''n'' in lowest terms with ''z'' > 0, then the infinitely many choices of log ''w'' yield only ''n'' different values for ''w''<sup>''z''</sup>; these values are the ''n'' complex solutions ''s'' to the equation ''s''<sup>''n''</sup> = ''w''<sup>''m''</sup>.
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| If ''z'' is an [[irrational number]], then the infinitely many choices of log ''w'' lead to infinitely many distinct values for ''w''<sup>''z''</sup>.
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| The computation of complex powers is facilitated by converting the base ''w'' to [[polar form]], as described in detail [[#Computing complex powers|below]].
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| A similar construction is employed in [[Quaternion#Exponential, logarithm, and power|quaternions]].
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| ===Complex roots of unity===
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| {{Main|Root of unity}}
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| [[File:One3Root.svg|thumb|right|The three 3rd roots of 1]]
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| A complex number ''w'' such that ''w''<sup>''n''</sup> = 1 for a positive integer ''n'' is an ''' ''n''th root of unity'''. Geometrically, the ''n''th roots of unity lie on the unit circle of the complex plane at the vertices of a regular ''n''-gon with one vertex on the real number 1.
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| If ''w''<sup>''n''</sup> = 1 but ''w''<sup>''k''</sup> ≠ 1 for all natural numbers ''k'' such that 0 < ''k'' < ''n'', then ''w'' is called a '''primitive ''n''th root of unity.''' The negative unit −1 is the only primitive square root of unity. The [[imaginary unit]] ''i'' is one of the two primitive 4-th roots of unity; the other one is −''i''.
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| The number ''e''<sup>{{sfrac|2''πi''|''n''}}</sup> is the primitive ''n''th root of unity with the smallest positive [[complex argument]]. (It is sometimes called the '''principal ''n''th root of unity''', although this terminology is not universal and should not be confused with the [[principal value]] of <sup>''n''</sup>√<span style="text-decoration:overline">1</span>, which is 1.<ref>This definition of a principal root of unity can be found in:
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| *{{cite book | title = Introduction to Algorithms | edition = second | author = Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein | publisher = MIT Press | year = 2001 | isbn = 0-262-03293-7}} [http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html Online resource]
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| *{{cite book | title = Difference Equations: From Rabbits to Chaos | edition = Undergraduate Texts in Mathematics | author = Paul Cull, Mary Flahive, and Robby Robson | year = 2005 | publisher = Springer | isbn = 0-387-23234-6 }} Defined on page 351, available on Google books.
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| * "[http://mathworld.wolfram.com/PrincipalRootofUnity.html Principal root of unity]", MathWorld.</ref>)
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| The other ''n''th roots of unity are given by
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| :<math>\left( e^{ \frac{2}{n} \pi i } \right) ^k = e^{ \frac{2}{n} \pi i k }</math>
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| for 2 ≤ ''k'' ≤ ''n''.
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| ===Roots of arbitrary complex numbers===
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| Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power ''w<sup>q</sup>'' in the important special case where ''q'' = 1/''n'' and ''n'' is a positive integer. These are the '''''n''th roots''' of ''w''; they are solutions of the equation ''z<sup>n</sup>'' = ''w''. As with real roots, a second root is also called a square root and a third root is also called a cube root.
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| It is conventional in mathematics to define ''w''<sup>1/''n''</sup> as the principal value of the root. If ''w'' is a positive real number, it is also conventional to select a positive real number as the principal value of the root ''w''<sup>1/''n''</sup>. For general complex numbers, the ''n''th root with the smallest argument is often selected as the principal value of the ''n''th root operation, as with principal values of roots of unity.
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| The set of ''n''th roots of a complex number ''w'' is obtained by multiplying the principal value ''w''<sup>1/''n''</sup> by each of the ''n''th roots of unity. For example, the fourth roots of 16 are 2, −2, 2''i'', and −2''i'', because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, ''i'', and −''i''.
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| ===Computing complex powers===
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| <!-- {{main|Polar coordinate system}} This is not the MAIN article on complex powers -->
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| It is often easier to compute complex powers by writing the number to be exponentiated in [[Principal argument|polar form]]. Every complex number ''z'' can be written in the polar form
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| :<math>z = re^{i\theta} = e^{\ln(r) + i\theta}</math>
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| where ''r'' is a nonnegative real number and θ is the (real) [[complex argument|argument]] of ''z''. The polar form has a simple geometric interpretation: if a complex number ''u'' + ''iv'' is thought of as representing a point (''u'', ''v'') in the [[complex plane]] using [[Cartesian coordinate system|Cartesian coordinates]], then (''r'', θ) is the same point in [[polar coordinates]]. That is, ''r'' is the "radius" ''r''<sup>2</sup> = ''u''<sup>2</sup> + ''v''<sup>2</sup> and θ is the "angle" θ = [[atan2]](''v'', ''u''). The polar angle θ is ambiguous since any integer multiple of 2π could be added to θ without changing the location of the point. Each choice of θ gives in general a different possible value of the power. A [[branch cut]] can be used to choose a specific value. The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number.
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| In order to compute the complex power ''w''<sup>''z''</sup>, write ''w'' in polar form:
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| :<math>w = r e^{i\theta}</math>
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| Then
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| :<math>\log w = \log r + i \theta</math>
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| and thus
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| :<math>w^z = e^{z \log w} = e^{z(\log r + i\theta)}</math>
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| If ''z'' is decomposed as ''c'' + ''di'', then the formula for ''w''<sup>''z''</sup> can be written more explicitly as
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| :<math>\left( r^c e^{-d\theta} \right) e^{i (d \log r + c\theta)} = \left( r^c e^{-d\theta} \right) \left[ \cos(d \log r + c\theta) + i \sin(d \log r + c\theta) \right]</math><!-- e^{c \log r - d\theta + i (d \log r + c\theta)} -->
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| This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form. It is shown here both in polar form and in Cartesian form (via Euler's identity).
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| The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute ''i''<sup>''i''</sup>, write ''i'' in polar and Cartesian forms:
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| :<math>\begin{align}
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| i &= 1 \cdot e^{\frac{1}{2} i \pi} \\
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| i &= 0 + 1i
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| \end{align}</math>
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| Then the formula above, with ''r'' = 1, θ = {{sfrac|π|2}}, ''c'' = 0, and ''d'' = 1, yields:
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| :<math>i^i = \left( 1^0 e^{-\frac{1}{2}\pi} \right) e^{i \left[1 \cdot \log 1 + 0 \cdot \frac{1}{2}\pi \right]} = e^{-\frac{1}{2}\pi} \approx 0.2079</math>
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| Similarly, to find (−2)<sup>3 + 4''i''</sup>, compute the polar form of −2,
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| :<math>-2 = 2e^{i \pi}</math>
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| and use the formula above to compute
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| :<math>(-2)^{3 + 4i} = \left( 2^3 e^{-4\pi} \right) e^{i[4\log(2) + 3\pi]} \approx (2.602 - 1.006 i) \cdot 10^{-5}</math>
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| The value of a complex power depends on the branch used. For example, if the polar form ''i'' = 1''e''<sup>{{sfrac|5''πi''|2}}</sup> is used to compute ''i'' <sup>''i''</sup>, the power is found to be ''e''<sup>−{{sfrac|5''π''|2}}</sup>; the principal value of ''i'' <sup>''i''</sup>, computed above, is ''e''<sup>−{{sfrac|π|2}}</sup>. The set of all possible values for ''i'' <sup>''i''</sup> is given by:<ref>[http://www.cut-the-knot.org/do_you_know/complex.shtml Complex number to a complex power may be real] at Cut The Knot gives some references to ''i''<sup>''i''</sup></ref>
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| :<math>\begin{align}
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| i &= 1 \cdot e^{\frac{1}{2} i\pi + i 2 \pi k} \big| k \isin \mathbb{Z} \\
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| i^i &= e^{i \left(\frac{1}{2} i\pi + i 2 \pi k\right)} \\
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| &= e^{-\left(\frac{1}{2} \pi + 2 \pi k\right)}
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| \end{align}</math>
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| So there is an infinity of values which are possible candidates for the value of ''i''<sup>''i''</sup>, one for each integer ''k''. All of them have a zero imaginary part so one can say ''i''<sup>''i''</sup> has an infinity of valid real values.
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| ===Failure of power and logarithm identities===
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| 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:
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| * The identity log(''b''<sup>''x''</sup>) = ''x'' · log ''b'' holds whenever ''b'' is a positive real number and ''x'' is a real number. But for the [[principal branch]] of the complex logarithm one has
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| *:: <math> i\pi = \log(-1) = \log\left[(-i)^2\right] \neq 2\log(-i) = 2\left(-\frac{i\pi}{2}\right) = -i\pi</math>
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| *: 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:
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| *:: <math>\log(w^z) \equiv z \cdot \log(w) \pmod{2 \pi i}</math>
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| *: This identity does not hold even when considering log as a multivalued function. The possible values of log(''w''<sup>''z''</sup>) contain those of ''z'' · log ''w'' as a subset. Using Log(''w'') for the principal value of log(''w'') and ''m'', ''n'' as any integers the possible values of both sides are:
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| *:: <math>\begin{align}
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| \left\{\log(w^z)\right\} &= \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n + 2 \pi i m \right\} \\
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| \left\{z \cdot \log(w)\right\} &= \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n \right\}
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| \end{align}</math>
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| * The identities (''bc'')<sup>''x''</sup> = ''b''<sup>''x''</sup>''c''<sup>''x''</sup> and (''b''/''c'')<sup>''x''</sup> = ''b''<sup>''x''</sup>/''c''<sup>''x''</sup> are valid when ''b'' and ''c'' are positive real numbers and ''x'' is a real number. But a calculation using principal branches shows that
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| *:: <math>1 = (-1\times -1)^\frac{1}{2} \not = (-1)^\frac{1}{2}(-1)^\frac{1}{2} = -1</math>
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| *: and
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| *::<math>i = (-1)^\frac{1}{2} = \left (\frac{1}{-1}\right )^\frac{1}{2} \not = \frac{1^\frac{1}{2}}{(-1)^\frac{1}{2}} = \frac{1}{i} = -i</math>
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| *: On the other hand, when ''x'' is an integer, the identities are valid for all nonzero complex numbers.
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| *: If exponentiation is considered as a multivalued function then the possible values of (−1×−1)<sup>1/2</sup> are {1, −1}. The identity holds but saying {1} = {(−1×−1)<sup>1/2</sup>} is wrong.
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| * The identity (e<sup>''x''</sup>)<sup>''y''</sup> = e<sup>''xy''</sup> holds for real numbers ''x'' and ''y'', but assuming its truth for complex numbers leads to the following [[Mathematical fallacy|paradox]], discovered in 1827 by [[Thomas Clausen (mathematician)|Clausen]]:<ref name="Clausen1827">{{cite journal |author=Steiner J, Clausen T, Abel NH |title=Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=270662 |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |volume=2 |year=1827 |pages=286–287}}</ref>
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| *: For any integer ''n'', we have:
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| *:#<math>e^{1 + 2 \pi i n} = e^{1} e^{2 \pi i n} = e \cdot 1 = e</math>
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| *:#<math>\left( e^{1+2\pi i n} \right)^{1 + 2 \pi i n} = e</math>
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| *:#<math>e^{1 + 4 \pi i n - 4 \pi^{2} n^{2}} = e</math>
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| *:#<math>e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e</math>
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| *:#<math>e^{-4 \pi^2 n^2} = 1</math>
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| *: but this is false when the integer ''n'' is nonzero.
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| *: There are a number of problems in the reasoning:
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| *: The major error is that changing the order of exponentiation in going from line two to three changes what the principal value chosen will be.
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| *: From the multi-valued point of view, the first error occurs even sooner. Implicit in the first line is that ''e'' is a real number, whereas the result of ''e''<sup>1+2π''in''</sup> is a complex number better represented as ''e''+0''i''. Substituting the complex number for the real on the second line makes the power have multiple possible values. Changing the order of exponentiation from lines two to three also affects how many possible values the result can have. <math>\scriptstyle (e^z)^w \;\ne\; e^{z w}</math>, but rather <math>\scriptstyle (e^z)^w \;=\; e^{(z \,+\, 2\pi i n) w}</math> multivalued over integers ''n''.
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| ==Zero to the power of zero==
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| [[Image:X^y.png|right|thumb|300px|Plot of {{nowrap|1=''z'' = ''x''<sup>''y''</sup>}}. The red curves (with ''z'' constant) yield different limits as (''x'',''y'') approaches (0,0). The green curves (of finite constant slope, {{nowrap|1=''y'' = ''ax''}}) all yield a limit of 1.]]
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| ===For discrete exponents===
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| There are many widely used formulas, having terms involving [[natural number|natural-number]] exponents, that require 0<sup>0</sup> to be evaluated to 1.
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| For example:
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| *Regarding ''b''<sup>0</sup> as an [[empty product]] assigns it the value 1, even when ''b'' = 0.
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| *The [[#Combinatorial interpretation|combinatorial interpretation]] of 0<sup>0</sup> is the number of [[empty tuple]]s of elements from the empty set. There is exactly one empty tuple.
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| *Equivalently, the [[#Exponentiation over sets|set-theoretic interpretation]] of 0<sup>0</sup> is the number of functions from the empty set to the empty set. There is exactly one such function, the [[empty function]].<ref name="Bourbaki">N. Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.</ref>
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| *The notation <math>\scriptstyle \sum a_nx^n</math> for [[polynomial]]s and [[power series]] rely on defining 0<sup>0</sup> = 1. Identities like <math>\scriptstyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n</math> and <math>\scriptstyle e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}</math> and the [[binomial theorem]] <math>\scriptstyle (1 + x)^n = \sum_{k = 0}^n \binom{n}{k} x^k</math> are not valid for {{nowrap|1=''x'' = 0}} unless {{nowrap|1=0<sup>0</sup> = 1}}.<ref>"Some textbooks leave the quantity 0<sup>0</sup> undefined, because the functions ''x''<sup>0</sup> and 0<sup>''x''</sup> have different limiting values when ''x'' decreases to 0. But this is a mistake. We must define {{nowrap|1=''x''<sup>0</sup> = 1}}, for all ''x'', if the binomial theorem is to be valid when {{nowrap|1=''x'' = 0}}, {{nowrap|1=''y'' = 0}}, and/or {{nowrap|1=''x'' = −''y''}}. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0<sup>''x''</sup> is quite unimportant".{{cite book|title=[[Concrete Mathematics]]|edition=1st|publisher=Addison Wesley Longman Publishing Co|date=1989-01-05|isbn=0-201-14236-8|author=[[Ronald Graham]], [[Donald Knuth]], and [[Oren Patashnik]]|page=162|chapter=Binomial coefficients}}</ref>
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| *In [[differential calculus]], the [[power rule]] <math>\scriptstyle \frac{d}{dx} x^n = nx^{n-1}</math> is not valid for {{nowrap|1=''n'' = 1}} at {{nowrap|1=''x'' = 0}} unless {{nowrap|1=0<sup>0</sup> = 1}}.
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| However, not all sources define 0<sup>0</sup> to be 1, particularly in the context of continuous exponents where the exponent could be viewed as a limit.
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| | |
| ===Continuous exponents===
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| When 0<sup>0</sup> arises as a [[limit of a function|limit]] of the form <math>\scriptstyle \lim_{x\rarr 0} f(x)^{g(x)}</math>, it must be handled as an [[indeterminate form]].
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| *Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.<ref>{{cite book|first=S. C.|last=Malik |coauthors=Savita Arora|year=1992|title=Mathematical Analysis|page=223|isbn=978-81-224-0323-7|quote=In general the limit of φ(''x'')/ψ(''x'') when ''x''=''a'' in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. But what happens when both limits are zero? The division (0/0) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are ∞/∞ 0 × ∞, ∞ − ∞, 0<sup>0</sup>, 1<sup>∞</sup> and ∞<sup>0</sup>.|publisher=Wiley|location=New York}}</ref> In fact, when ''f''(''t'') and ''g''(''t'') are real-valued functions both approaching 0 (as ''t'' approaches a real number or ±∞), with ''f''(''t'') > 0, the function ''f''(''t'')<sup>''g''(''t'')</sup> need not approach 1; depending on ''f'' and ''g'', the limit of ''f''(''t'')<sup>''g''(''t'')</sup> can be any nonnegative real number or +∞, or it can be [[undefined (mathematics)|undefined]]. For example, the functions below are of the form ''f''(''t'')<sup>''g''(''t'')</sup> with ''f''(''t''),''g''(''t'') → 0 as [[one-sided limit|''t'' → 0<sup>+</sup>]], but the limits are different:
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| ::<math> \lim_{t \to 0^+} {t}^{t} = 1, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^t = 0, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^{-t} = +\infty, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t}}\right)^{at} = e^{-a}</math>.
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| | |
| :So 0<sup>0</sup> is an indeterminate form. This behavior shows that the two-variable function ''x''<sup>''y''</sup>, though continuous on the set {(''x'',''y''): ''x'' > 0}, cannot be extended to a [[continuous function]] on any set containing (0,0), no matter how 0<sup>0</sup> is defined.<ref>{{cite journal|author=L. J. Paige|title=A note on indeterminate forms|jstor=2307224|journal=American Mathematical Monthly|volume=61|issue=3|date=March 1954|pages=189–190|doi=10.2307/2307224}}</ref> However, under certain conditions, such as when ''f'' and ''g'' are both [[analytic functions]] and ''f'' is positive on the open interval (0,''b'') for some positive ''b'', the limit approaching from the right is always 1.<ref>[http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ sci.math FAQ: What is 0^0?]</ref><ref>{{Cite journal | doi = 10.2307/2689754| jstor = 2689754 | pages = 41–42 | title = The Indeterminate Form 0<sup>0</sup> | journal = [[Mathematics Magazine]] | volume = 50 | issue = 1 | publisher = [[Mathematical Association of America]] | year = 1977 | author1 = Rotando | first1 = Louis M. | author2 = Korn | first2 = Henry }}</ref><ref>{{Cite journal | doi = 10.2307/3595845 | jstor = 3595845 | pages = 55–56 | title = On the Indeterminate Form 0<sup>0</sup> | journal = [[The College Mathematics Journal]] | volume = 34 | issue = 1 | publisher = [[Mathematical Association of America]] | year = 2003 | author1 = Lipkin | first1 = Leonard J.}}</ref>
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| *In the [[complex domain]], the function ''z''<sup>''w''</sup> is defined for nonzero ''z'' by choosing a [[complex logarithm#Branches of the complex logarithm|branch]] of log ''z'' and setting ''z''<sup>''w''</sup> := ''e''<sup>''w'' log ''z''</sup>, but there is no branch of log ''z'' defined at ''z'' = 0, let alone in a neighborhood of 0.<ref>"Since ln(0) does not exist, 0<sup>''z''</sup> is undefined. For Re(''z'') > 0, we define it arbitrarily as 0." George F. Carrier, Max Krook and Carl E. Pearson, ''Functions of a Complex Variable: Theory and Technique '', 2005, p. 15</ref><ref>"For ''z''=0, ''w''≠0, we define 0<sup>w</sup> = 0, while 0<sup>0</sup> is not defined." Mario Gonzalez, ''Classical Complex Analysis'', Chapman & Hall, 1991, p. 56.</ref><ref>"... Let's start at ''x'' = 0. Here ''x''<sup>''x''</sup> is undefined." Mark D. Meyerson, The ''x''<sup>''x''</sup> Spindle, ''Mathematics Magazine'' '''69''', no. 3 (June 1996), 198-206.</ref>
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| ===History of differing points of view===
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| The debate over the definition of 0<sup>0</sup> has been going on at least since the early 19th century. At that time, most mathematicians agreed that 0<sup>0</sup> = 1, until in 1821 [[Cauchy]]<ref>Augustin-Louis Cauchy, ''Cours d'Analyse de l'École Royale Polytechnique'' (1821). In his ''Oeuvres Complètes'', series 2, volume 3.</ref> listed 0<sup>0</sup> along with expressions like {{sfrac|0|0}} in a table of indeterminate forms. In the 1830s Libri<ref>Guillaume Libri, Note sur les valeurs de la fonction 0<sup>0<sup>x</sup></sup>, ''[[Crelle's Journal|Journal für die reine und angewandte Mathematik]]'' '''6''' (1830), 67–72.</ref><ref>Guillaume Libri, Mémoire sur les fonctions discontinues, ''Journal für die reine und angewandte Mathematik'' '''10''' (1833), 303–316.</ref> published an unconvincing argument for 0<sup>0</sup> = 1, and [[August Ferdinand Möbius|Möbius]]<ref>A. F. Möbius, Beweis der Gleichung 0<sup>0</sup> = 1, nach [[Johann Friedrich Pfaff|J. F. Pfaff]], ''Journal für die reine und angewandte Mathematik'' '''12''' (1834), 134–136.</ref> sided with him, erroneously claiming that <math>\scriptstyle \lim_{t \to 0^+} f(t)^{g(t)} \;=\; 1</math> whenever <math>\scriptstyle \lim_{t \to 0^+} f(t) \;=\; \lim_{t \to 0^+} g(t) \;=\; 0</math>. A commentator who signed his name simply as "S" provided the counterexample of (''e''<sup>−1/''t''</sup>)<sup>''t''</sup>, and this quieted the debate for some time. More historical details can be found in Knuth (1992).<ref name="Knuth1992" />
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| More recent authors interpret the situation above in different ways:
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| * Some argue that the best value for 0<sup>0</sup> depends on context, and hence that [[defined and undefined|defining]] it once and for all is problematic.<ref>Examples include Edwards and Penny (1994). ''Calculus'', 4th ed,, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). ''Algebra Two.'' Addison-Wesley, p. 32.</ref> According to Benson (1999), "The choice whether to define 0<sup>0</sup> is based on convenience, not on correctness."<ref>Donald C. Benson, ''The Moment of Proof : Mathematical Epiphanies.'' New York Oxford University Press (UK), 1999. ISBN 978-0-19-511721-9</ref>
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| * Others argue that 0<sup>0</sup> should be defined as 1. [[Donald Knuth|Knuth]] (1992) contends strongly that 0<sup>0</sup> "''has'' to be 1", drawing a distinction between the ''value'' 0<sup>0</sup>, which should equal 1 as advocated by Libri, and the ''limiting form'' 0<sup>0</sup> (an abbreviation for a limit of <math>\scriptstyle f(x)^{g(x)}</math> where <math>\scriptstyle f(x), g(x) \to 0</math>), which is necessarily an [[indeterminate form]] as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."<ref name="Knuth1992">Donald E. Knuth, Two notes on notation, ''[[American Mathematical Monthly|Amer. Math. Monthly]]'' '''99''' no. 5 (May 1992), 403–422 ([http://arxiv.org/abs/math/9205211 arXiv:math/9205211 [math.HO]]).</ref>
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| ===Treatment on computers===
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| | |
| ====IEEE floating point standard====
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| The [[IEEE 754-2008]] floating point standard is used in the design of most floating point libraries. It recommends a number of different functions for computing a power:<ref>{{cite book|title=Handbook of Floating-Point Arithmetic|publisher=Birkhäuser Boston|year=2009|isbn=978-0-8176-4704-9|page=216|author9=Jean-Michel Muller et al}}</ref>
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| *<tt>pow</tt> treats 0<sup>0</sup> as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for <tt>pown</tt>, otherwise the result is as for <tt>powr</tt> (except for some exceptional cases).
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| *<tt>pown</tt> treats 0<sup>0</sup> as 1. The power must be an exact integer. The value is defined for negative bases; e.g., <tt>pown(−3,5)</tt> is −243.
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| *<tt>powr</tt> treats 0<sup>0</sup> as [[NaN]] (Not-a-Number – undefined). The value is also NaN for cases like <tt>powr(−3,2)</tt> where the base is less than zero. The value is defined by ''e''<sup>power×log(base)</sup>.
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| ====Programming languages====
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| Most programming language with a power function are implemented using the IEEE <tt>pow</tt> function and therefore evaluate 0<sup>0</sup> as 1. The later C<ref>{{cite journal|title=Rationale for International Standard—Programming Languages—C|version=Revision 5.10|date=April 2003|author=John Benito|url=http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf|page=182|format=PDF}}</ref> and C++ standards describe this as the [[normative]] behaviour. The [[Java (programming language)|Java]] standard<ref>{{cite web|url=http://download.oracle.com/javase/1.4.2/docs/api/java/lang/Math.html#pow%28double,%20double%29 |title=Math (Java 2 Platform SE 1.4.2) pow |publisher=Oracle}}</ref> mandates this behavior. The [[.NET Framework]] [[method (computer science)|method]] <code>System.Math.Pow</code> also treats 0<sup>0</sup> as 1.<ref>{{cite web |url=http://msdn.microsoft.com/en-us/library/system.math.pow.aspx |title=.NET Framework Class Library Math.Pow Method |publisher=Microsoft}}</ref>
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| ====Mathematics software====
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| *[[Sage (mathematics software)|Sage]] simplifies ''b''<sup>0</sup> to 1, even if no constraints are placed on ''b''.<ref>{{cite web |url=http://sagenb.org/home/pub/2433/ | title=Sage worksheet calculating x^0 |publisher=Jason Grout}}</ref> It takes 0<sup>0</sup> to be 1, but does not simplify 0<sup>''x''</sup> for other ''x''.
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| *[[Maple (software)|Maple]] simplifies ''b''<sup>0</sup> to 1 even if no constraints are placed on ''b'', and evaluates 0<sup>0</sup> to 1. For ''x''>0, it simplifies 0<sup>''x''</sup> to 0.{{Citation needed|date=July 2010}}
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| *[[Macsyma]] also simplifies ''b''<sup>0</sup> to 1 even if no constraints are placed on ''b'', but issues an error for 0<sup>0</sup>. For ''x''>0, it simplifies 0<sup>''x''</sup> to 0. {{Citation needed|date=July 2010}}
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| *[[Mathematica]] and [[Wolfram Alpha]] simplify ''b''<sup>0</sup> into 1, even if no constraints are placed on ''b''.<ref>{{cite web |url=http://www.wolframalpha.com/input/?i=a^0 | title=Wolfram Alpha calculates b^0 |publisher=Wolfram Alpha LLC, accessed July 24, 2011}}</ref> While Mathematica does not simplify 0<sup>''x''</sup>, Wolfram Alpha returns two results, 0 and indeterminate.<ref>{{cite web |url=http://www.wolframalpha.com/input/?i=0^a | title=Wolfram Alpha calculates 0^a |publisher=Wolfram Alpha LLC, accessed July 24, 2011}}</ref> Both Mathematica and Wolfram Alpha take 0<sup>0</sup> to be an [[indeterminate form]].<ref>{{cite web |url=http://www.wolframalpha.com/input/?i=0^0 | title=Wolfram Alpha calculates 0^0 |publisher=Wolfram Alpha LLC, accessed July 24, 2011}}</ref>
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| ==Limits of powers==
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| The section [[#Zero to the power of zero|zero to the power of zero]] gives a number of examples of limits which are of the [[indeterminate form]] 0<sup>0</sup>. The limits in these examples exist, but have different values, showing that the two-variable function ''x''<sup>''y''</sup> has no limit at the point (0,0). One may ask at what points this function does have a limit.
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| More precisely, consider the function ''f''(''x'',''y'') = ''x''<sup>''y''</sup> defined on ''D'' = {(''x'',''y'') ∈ '''R'''<sup>2</sup> : ''x'' > 0}. Then ''D'' can be viewed as a subset of {{overline|'''R'''}}<sup>2</sup> (that is, the set of all pairs (''x'',''y'') with ''x'',''y'' belonging to the [[extended real number line]] {{overline|'''R'''}} = [−∞, +∞], endowed with the [[product topology]]), which will contain the points at which the function ''f'' has a limit.
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| In fact, ''f'' has a limit at all [[accumulation point]]s of ''D'', except for (0,0), (+∞,0), (1,+∞) and (1,−∞).<ref>N. Bourbaki, ''Topologie générale'', V.4.2.</ref> Accordingly, this allows one to define the powers ''x''<sup>''y''</sup> by continuity whenever 0 ≤ ''x'' ≤ +∞, −∞ ≤ y ≤ +∞, except for 0<sup>0</sup>, (+∞)<sup>0</sup>, 1<sup>+∞</sup> and 1<sup>−∞</sup>, which remain indeterminate forms.
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| Under this definition by continuity, we obtain:
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| * ''x''<sup>+∞</sup> = +∞ and ''x''<sup>−∞</sup> = 0, when 1 < ''x'' ≤ +∞.
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| * ''x''<sup>+∞</sup> = 0 and ''x''<sup>−∞</sup> = +∞, when 0 ≤ ''x'' < 1.
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| * 0<sup>''y''</sup> = 0 and (+∞)<sup>''y''</sup> = +∞, when 0 < ''y'' ≤ +∞.
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| * 0<sup>''y''</sup> = +∞ and (+∞)<sup>''y''</sup> = 0, when −∞ ≤ ''y'' < 0.
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| These powers are obtained by taking limits of ''x''<sup>''y''</sup> for ''positive'' values of ''x''. This method does not permit a definition of ''x''<sup>''y''</sup> when ''x'' < 0, since pairs (''x'',''y'') with ''x'' < 0 are not accumulation points of ''D''.
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| On the other hand, when ''n'' is an integer, the power ''x''<sup>''n''</sup> is already meaningful for all values of ''x'', including negative ones. This may make the definition 0<sup>''n''</sup> = +∞ obtained above for negative ''n'' problematic when ''n'' is odd, since in this case ''x''<sup>''n''</sup> → +∞ as ''x'' tends to 0 through positive values, but not negative ones.
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| ==Efficient computation with integer exponents==
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| The simplest method of computing ''b''<sup>''n''</sup> requires {{nowrap|''n'' − 1}} multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2<sup>100</sup>, note that {{nowrap|1=100 = 64 + 32 + 4}}. Compute the following in order:
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| # 2<sup>2</sup> = 4
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| # (2<sup>2</sup>)<sup>2</sup> = 2<sup>4</sup> = 16
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| # (2<sup>4</sup>)<sup>2</sup> = 2<sup>8</sup> = 256
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| # (2<sup>8</sup>)<sup>2</sup> = 2<sup>16</sup> = 65,536
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| # (2<sup>16</sup>)<sup>2</sup> = 2<sup>32</sup> = 4,294,967,296
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| # (2<sup>32</sup>)<sup>2</sup> = 2<sup>64</sup> = 18,446,744,073,709,551,616
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| # 2<sup>64</sup> 2<sup>32</sup> 2<sup>4</sup> = 2<sup>100</sup> = 1,267,650,600,228,229,401,496,703,205,376
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| This series of steps only requires 8 multiplication operations instead of 99 (since the last product above takes 2 multiplications).
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| In general, the number of multiplication operations required to compute
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| ''b''<sup>''n''</sup> can be reduced to [[asymptotic notation|Θ]](log ''n'') by using [[exponentiation by squaring]] or (more generally) [[addition-chain exponentiation]]. Finding the ''minimal'' sequence of multiplications (the minimal-length addition chain for the exponent) for ''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 doi|10.1006/jagm.1997.0913}}</ref>
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| ==Exponential notation for function names==
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| Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated [[function composition]] rather than repeated multiplication. Thus ''f''<sup> 3</sup>(''x'') may mean ''f''(''f''(''f''(''x''))); in particular, ''f''<sup> −1</sup>(''x'') usually denotes the [[inverse function]] of ''f''. [[Iterated function]]s are of interest in the study of [[fractal]]s and [[dynamical systems]]. [[Babbage]] was the first to study the problem of finding a [[functional square root]] ''f''<sup> 1/2</sup>(''x'').
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| However, for historical reasons, a special syntax applies to the [[trigonometric functions]]: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. That is, sin<sup>2</sup>''x'' is just a shorthand way to write (sin ''x'')<sup>2</sup> without using parentheses, whereas sin<sup>−1</sup>''x'' refers to the inverse function of the [[sine]], also called arcsin ''x''. There is no need for a shorthand for the reciprocals of trigonometric functions since each has its own name and abbreviation; for example, 1/(sin ''x'') = (sin ''x'')<sup>−1</sup> = csc ''x''. A similar convention applies to logarithms, where log<sup>2</sup>''x'' usually means (log ''x'')<sup>2</sup>, not log log ''x''.
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| ==Generalizations==
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| ===In abstract algebra===
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| Exponentiation for integer exponents can be defined for quite general structures in [[abstract algebra]].
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| Let ''X'' be a [[Set (mathematics)|set]] with a [[power-associative]] [[binary operation]] which is written multiplicatively. Then ''x''<sup>''n''</sup> is defined for any element ''x'' of ''X'' and any nonzero [[natural number]] ''n'' as the product of ''n'' copies of ''x'', which is recursively defined by
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| :<math>\begin{align}
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| x^1 &= x \\
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| x^n &= x^{n-1}x \quad\hbox{for }n>1
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| \end{align}</math>
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| One has the following properties
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| :<math>\begin{align}
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| (x^i x^j) x^k &= x^i (x^j x^k) \quad\text{(power-associative property)} \\
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| x^{m+n} &= x^m x^n \\
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| (x^m)^n &= x^{mn}
| |
| \end{align}</math>
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| If the operation has a two-sided [[identity element]] 1 (often denoted by ''e''), then ''x''<sup>0</sup> is defined to be equal to 1 for any ''x''.
| |
| :<math>\begin{align}
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| x1 &= 1x = x \quad\text{(two-sided identity)} \\
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| x^0 &= 1
| |
| \end{align}</math>
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| | |
| If the operation also has two-sided [[inverse element|inverses]], and multiplication is associative then the [[Magma (algebra)|magma]] is a [[group (mathematics)|group]]. The inverse of ''x'' can be denoted by ''x''<sup>−1</sup> and follows all the usual rules for exponents.
| |
| :<math>\begin{align}
| |
| x x^{-1} &= x^{-1} x = 1 \quad\text{(two-sided inverse)} \\
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| (x y) z &= x (y z) \quad\text{(associative)} \\
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| x^{-n} &= \left(x^{-1}\right)^n \\
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| x^{m-n} &= x^m x^{-n}
| |
| \end{align}</math>
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| | |
| If the multiplication operation is [[commutative]] (as for instance in [[abelian group]]s), then the following holds:
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| : <math>(xy)^n = x^n y^n </math>
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| If the binary operation is written additively, as it often is for [[abelian groups]], then "exponentiation is repeated multiplication" can be reinterpreted as "[[multiplication]] is repeated [[addition]]". Thus, each of the laws of exponentiation above has an [[analogy|analogue]] among laws of multiplication.
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| When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, ''x''<sup>∗''n''</sup> is ''x'' ∗ ··· ∗ ''x'', while ''x''<sup>#''n''</sup> is ''x'' # ··· # ''x'', whatever the operations ∗ and # might be.
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| Superscript notation is also used, especially in [[group theory]], to indicate [[Conjugacy class|conjugation]]. That is, ''g''<sup>''h''</sup> = ''h''<sup>−1</sup>''gh'', where ''g'' and ''h'' are elements of some [[group (math)|group]]. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A [[quandle]] is an [[algebraic structure]] in which these laws of conjugation play a central role.
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| ==={{Anchor|Exponentiation over sets}}Over sets===
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| {{Main|Cartesian product}}
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| If ''n'' is a natural number and ''A'' is an arbitrary set, the expression ''A''<sup>''n''</sup> is often used to denote the set of ordered [[n-tuple|''n''-tuple]]s of elements of ''A''. This is equivalent to letting ''A''<sup>''n''</sup> denote the set of functions from the set {0, 1, 2, …, ''n''−1} to the set ''A''; the ''n''-tuple (''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, …, a<sub>''n''−1</sub>) represents the function that sends ''i'' to ''a''<sub>''i''</sub>.
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| For an infinite [[cardinal number]] κ and a set ''A'', the notation ''A''<sup>κ</sup> is also used to denote the set of all functions from a set of size κ to ''A''. This is sometimes written <sup>κ</sup>''A'' to distinguish it from cardinal exponentiation, defined below.
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| This generalized exponential can also be defined for operations on sets or for sets with extra [[Mathematical structure|structure]]. For example, in [[linear algebra]], it makes sense to index [[Direct sum of modules|direct sum]]s of [[vector space]]s over arbitrary index sets. That is, we can speak of
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| : <math>\bigoplus_{i \in \mathbb{N}} V_{i}</math>
| |
| | |
| where each ''V''<sub>''i''</sub> is a vector space.
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| Then if ''V''<sub>''i''</sub> = ''V'' for each ''i'', the resulting direct sum can be written in exponential notation as ''V''<sup>⊕'''N'''</sup>, or simply ''V''<sup>'''N'''</sup> with the understanding that the direct sum is the default. We can again replace the set '''N''' with a cardinal number ''n'' to get ''V''<sup>''n''</sup>, although without choosing a specific standard set with cardinality ''n'', this is defined only [[up to]] [[isomorphism]]. Taking ''V'' to be the [[field (algebra)|field]] '''R''' of [[real number]]s (thought of as a vector space over itself) and ''n'' to be some [[natural number]], we get the vector space that is most commonly studied in linear algebra, the [[Euclidean space]] '''R'''<sup>''n''</sup>.
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| If the base of the exponentiation operation is a set, the exponentiation operation is the [[Cartesian product]] unless otherwise stated. Since multiple Cartesian products produce an ''n''-[[tuple]], which can be represented by a function on a set of appropriate cardinality, ''S''<sup>''N''</sup> becomes simply the set of all [[Function (mathematics)|function]]s from ''N'' to ''S'' in this case:
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| : <math>S^N \equiv \{ f\colon N \to S \}</math>
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| This fits in with the exponentiation of cardinal numbers, in the sense that |''S''<sup>''N''</sup>| = |''S''|<sup>|''N''|</sup>, where |''X''| is the cardinality of ''X''. When "2" is defined as {0, 1}, we have |2<sup>''X''</sup>| = 2<sup>|''X''|</sup>, where 2<sup>''X''</sup>, usually denoted by '''P'''(''X''), is the [[power set]] of ''X''; each [[subset]] ''Y'' of ''X'' corresponds uniquely to a function on ''X'' taking the value 1 for ''x'' ∈ ''Y'' and 0 for ''x'' ∉ ''Y''.<!-- (This is where the term "power set" comes from. This needs a source -->
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| ===In category theory===
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| {{Main|Cartesian closed category}}
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| In a [[Cartesian closed category]], the [[exponential (category theory)|exponential]] operation can be used to raise an arbitrary object to the power of another object. This generalizes the [[Cartesian product]] in the category of sets. If 0 is an [[initial object]] in a Cartesian closed category, then the [[exponential object]] 0<sup>0</sup> is isomorphic to any terminal object 1.
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| ===Of cardinal and ordinal numbers===
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| {{Main|Cardinal number#Cardinal arithmetic|l1=cardinal arithmetic|ordinal arithmetic}}
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| In [[set theory]], there are exponential operations for [[cardinal number|cardinal]] and [[ordinal number]]s.
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| If κ and λ are cardinal numbers, the expression κ<sup>λ</sup> represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.<ref name="Bourbaki"/> If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality 8 = 2<sup>3</sup>.
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| Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a [[limit (mathematics)|limit]] process involving [[transfinite induction]].
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| ==Repeated exponentiation==
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| 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 [[tetration]]. Iterating tetration leads to another operation, and so on. 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 (3,3), the functions addition, multiplication, exponentiation, tetration yield 6, 9, 27, and 7,625,597,484,987 respectively.
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| ==In programming languages==
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| The superscript notation ''x''<sup>''y''</sup> is convenient in handwriting but inconvenient for [[typewriter]]s and [[computer terminal]]s that align the baselines of all characters on each line. Many [[programming language]]s have alternate ways of expressing exponentiation that do not use superscripts:
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| * <code>x ↑ y</code>: [[Algol programming language|Algol]], [[Commodore BASIC]]
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| * <code>x ^ y</code>: [[BASIC]], [[J programming language|J]], [[MATLAB]], [[R (programming language)|R]], [[Microsoft Excel]], [[TeX]] (and its derivatives), [[TI-BASIC]], [[bc programming language|bc]] (for integer exponents), [[Haskell (programming language)|Haskell]] (for nonnegative integer exponents), [[Lua (programming language)|Lua]], [[Active Server Pages|ASP]] and most [[computer algebra system]]s
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| * <code>x ^^ y</code>: Haskell (for fractional base, integer exponents), [[D (programming language)|D]]
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| * <code>x ** y</code>: [[Ada (programming language)|Ada]], [[Bash (Unix shell)|Bash]], [[COBOL]], [[Fortran]], [[FoxPro 2|FoxPro]], [[Gnuplot]], [[OCaml]], [[F Sharp (programming language)|F#]], [[Perl]], [[PL/I]], [[Python (programming language)|Python]], [[Rexx]], [[Ruby (programming language)|Ruby]], [[SAS programming language|SAS]], [[Seed7]], [[Tcl]], [[ABAP]], Haskell (for floating-point exponents), [[Turing (programming language)|Turing]], [[VHDL]]
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| * <code>pown x y</code>: F# (for integer base, integer exponent)
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| * <code>x⋆y</code>: [[APL (programming language)|APL]]
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| Many programming languages lack syntactic support for exponentiation, but provide library functions.
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| In Bash, C, C++, C#, Java, JavaScript, Perl, PHP, Python and Ruby, the symbol ^ represents bitwise [[XOR]]. In Pascal, it represents [[indirection]]. In OCaml and Standard ML, it represents string [[concatenation]].
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| ==History of the notation==
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| The term ''power'' was used by the [[Greek mathematics|Greek]] mathematician [[Euclid]] for the square of a line.<ref name=MacTutor/> [[Archimedes]] discovered and proved the law of exponents, 10<sup>a</sup> 10<sup>b</sup> = 10<sup>a+b</sup>, necessary to manipulate powers of 10.<ref>For further analysis see [[The Sand Reckoner]].</ref> In the 9th century, the Persian mathematician [[Muhammad ibn Mūsā al-Khwārizmī]] used the terms ''mal'' for a [[Square (algebra)|square]] and ''kab'' for a [[Cube (algebra)|cube]], which later [[Mathematics in medieval Islam|Islamic]] mathematicians represented in [[mathematical notation]] as ''m'' and ''k'', respectively, by the 15th century, as seen in the work of [[Abū al-Hasan ibn Alī al-Qalasādī]].<ref>{{MacTutor|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref>
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| [[Nicolas Chuquet]] used a form of exponential notation in the 15th century, which was later used by [[Henricus Grammateus]] and [[Michael Stifel]] in the 16th century. [[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> In the 16th century [[Robert Recorde]] used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and [[zenzizenzizenzic]] (eighth).<ref>{{Cite web|url=http://www.worldwidewords.org/weirdwords/ww-zen1.htm|title=Zenzizenzizenzic - the eighth power of a number|publisher=World Wide Words|first=Michael|last=Quinion|accessdate=2010-03-19}}</ref> ''Biquadrate'' has been used to refer to the fourth power as well.
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| Some mathematicians (e.g., [[Isaac Newton]]) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ''ax'' + ''bxx'' + ''cx''<sup>3</sup> + ''d''.
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| Another historical synonym, '''involution''',<ref>This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary [http://www.m-w.com/dictionary/involution]. The most recent usage in this sense cited by the OED is from 1806.</ref> is now rare and should not be confused with [[Involution (mathematics)|its more common meaning]].
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| ==List of whole-number exponentials==
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| {|class="wikitable"
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| !''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>
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| |-
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| |width="8%"|'''2''' || width="8%"|4|| width="7%"|8|| width="8%"|16|| width="9%"|32|| width="10%"|64|| width="11%"|128|| width="12%"|256|| width="12%"|512|| width="14%"|1,024
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| |-
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| |'''3''' ||9||27||81||243||729||2,187||6,561||19,683||59,049
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| |-
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| |'''4''' ||16||64||256||1,024||4,096||16,384||65,536||262,144||1,048,576
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| |-
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| |'''5''' ||25||125||625||3,125||15,625||78,125||390,625||1,953,125||9,765,625
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| |-
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| |'''6''' ||36||216||1,296||7,776||46,656||279,936||1,679,616||10,077,696||60,466,176
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| |-
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| |'''7''' ||49||343||2,401||16,807||117,649||823,543||5,764,801||40,353,607||282,475,249
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| |-
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| |'''8''' ||64||512||4,096||32,768||262,144||2,097,152||16,777,216||134,217,728||1,073,741,824
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| |-
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| |'''9''' ||81||729||6,561||59,049||531,441||4,782,969||43,046,721||387,420,489||3,486,784,401
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| |-
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| |'''10'''||100||1,000||10,000||100,000||1,000,000||10,000,000||100,000,000||1,000,000,000||10,000,000,000
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| |}
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| ==See also==
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| {{col-begin}}
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| {{col-break|width=33%}}
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| *[[Exponential decay]]
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| *[[Exponential growth]]
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| *[[List of exponential topics]]
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| {{col-break}}
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| {{Portal|Mathematics}}<!-- Located here after first col-break for better rendering -->
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| *[[Modular exponentiation]]
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| *[[Unicode subscripts and superscripts]]
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| {{col-end}}
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| ==References==
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| {{Reflist|30em}}
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| ==External links==
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| * [http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ sci.math FAQ: What is 0<sup>0</sup>?]
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| *{{planetmath reference|id=3948|title=Introducing 0th power}}
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| * [http://www.mathsisfun.com/algebra/exponent-laws.html Laws of Exponents] with derivation and examples
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| * [http://www.askamathematician.com/?p=4524 What does 0^0 (zero to the zeroth power) equal?] on AskAMathematician.com
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| {{Good article}}
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| [[Category:Exponentials]]
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| [[Category:Binary operations]]
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| {{Link FA|he}}
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| {{Link GA|th}}
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