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

$a^{m}\!\cdot\!a^{n}\;=\;a^{m+n},\quad(a^{m})^{n}\;=\;a^{mn}$

are . For the validity of the third well-known 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 $a^{-1}$ ):

$a^{0}\;:=\;e,\qquad a^{-n}\;:=\;(a^{-1})^{n}$

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

$a^{r_{1}},\,a^{r_{2}},\,\ldots$

where the limit of the rational number sequence $r_{1},\,r_{2},\,\ldots$ is $\varrho$ . The sequence $a^{r_{1}},\,a^{r_{2}},\,\ldots$ may be shown to be a Cauchy sequence.
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The last step were the imaginary (non-real complex) exponents $\mu$ , when also the base of the power may be other than a positive real number; the one gets the so-called general power.

Title	exponentiation
Canonical name	Exponentiation
Date of creation	2013-03-22 19:08:44
Last modified on	2013-03-22 19:08:44
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	9
Author	pahio (2872)
Entry type	Topic
Classification	msc 20-00
Related topic	ContinuityOfNaturalPower
Defines	power law
Defines	power of product