exponentiation

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In the entry general associativity, the notion of the power ${a}^{n}$ for elements $a$ of a set having an associative binary operation^{} “$\cdot $” and for positive integers $n$ as exponents^{} (http://planetmath.org/GeneralPower) was defined as a generalisation of the operation. Then the two power laws
$${a}^{m}\cdot {a}^{n}={a}^{m+n},{({a}^{m})}^{n}={a}^{mn}$$ are . For the validity of the third wellknown power law,
$${(a\cdot b)}^{n}={a}^{n}\cdot {b}^{n},$$ the law of power of product, the commutativity of the operation is needed.
Example. In the symmetric group^{} ${S}_{3}$, where the group operation^{} is not commutative^{}, we get different results from
$${[(123)(13)]}^{2}={(23)}^{2}=(1)$$ and
$${(123)}^{2}{(13)}^{2}=(132)(1)=(132)$$ (note that in these “products^{}”, which compositions of mappings, the latter “factor” acts first).

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Extending the power notion for zero and negative integer exponents requires the existence of http://planetmath.org/node/10539neutral and inverse elements ($e$ and ${a}^{1}$):
$${a}^{0}:=e,{a}^{n}:={({a}^{1})}^{n}$$ The two first power laws then remain in for all integer exponents, and if the operation is commutative, also the .
When the operation in question is the multiplication of real or complex numbers, the power notion may be extended for other than integer exponents.

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One step is to introduce fractional (http://planetmath.org/FractionalNumber) exponents by using roots (http://planetmath.org/NthRoot); see the fraction power.

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The following step would be the irrational exponents, which are in the power functions^{}. The irrational exponents are possible to introduce by utilizing the exponential function^{} and logarithms; another way would be to define ${a}^{\varrho}$ as limit of a sequence^{}
$${a}^{{r}_{1}},{a}^{{r}_{2}},\mathrm{\dots}$$ where the limit of the rational number sequence ${r}_{1},{r}_{2},\mathrm{\dots}$ is $\varrho $. The sequence ${a}^{{r}_{1}},{a}^{{r}_{2}},\mathrm{\dots}$ may be shown to be a Cauchy sequence.

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The last step were the imaginary (nonreal complex) exponents $\mu $, when also the base of the power may be other than a positive real number; the one gets the socalled general power.
Title  exponentiation 

Canonical name  Exponentiation 
Date of creation  20130322 19:08:44 
Last modified on  20130322 19:08:44 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  9 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 2000 
Related topic  ContinuityOfNaturalPower 
Defines  power law 
Defines  power of product 