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 (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 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).

• 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.

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

• 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.

• 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.