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exponentiation

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 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 valid. 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 mean compositions of mappings, the latter “factor” acts first).

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

One step is to introduce fractional exponents by using roots; see the fraction power.

The following step would be the irrational exponents, which are contained 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.

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.
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