Power

2020 Mathematics Subject Classification: Primary: 97Fxx [MSN][ZBL]

In its primary meaning (an integer and positive power), it is the product of a number of equal factors and is written in the form $a^n = a\cdots a$ ($n$ times), where $a$ is the base, $n$ the exponent and $a^n$ the power. The basic properties of powers are:

$$a^n\cdot a^m = a^{n+m},\quad (ab)^n = a^n b^n,\quad \frac{a^n}{a^m} = a^{n-m},\quad (a^n)^m = a^{nm}.$$ Further generalizations of the idea of a power include: zero powers: $a^0 = 1 $, negative powers: $a^{-n} = 1/a^n$ (when $a\ne 0$); fractional powers: $a^{n/m} = (a^{1/m})^n$, where $a^{1/m} = \sqrt[m]{a}$ (for $a>0$); and a power with an irrational exponent: $\def\a{\alpha}a^\a = \lim_{r_n\to\a} a^{r_n}$, where $r_n$ is an arbitrary sequence of rational numbers tending to $\a$. Powers with a complex base (see de Moivre formula) and powers with a complex base and complex exponent (by definition: $z^u = e^{u\mathrm{Ln} z}$) are also studied.