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(See also Exponential function and Exponential function, real)
 
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741901.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741902.png" /> times), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741903.png" /> is the base, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741904.png" /> the exponent and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741905.png" /> the power. The basic properties of powers are:
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{{MSC|97Fxx}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741906.png" /></td> </tr></table>
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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:
  
Further generalizations of the idea of a power include: zero powers: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741907.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741908.png" />); negative powers: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p0741909.png" />; fractional powers: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p07419010.png" />; and a power with an irrational exponent: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p07419011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p07419012.png" /> is an arbitrary sequence of rational numbers tending to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p07419013.png" />. Powers with a complex base (see [[De Moivre formula|de Moivre formula]]) and powers with a complex base and complex exponent (by definition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p07419014.png" />) are also studied.
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$$a^n\cdot a^m = a^{n+m},\quad (ab)^n = a^n b^n,\quad \frac{a^n}{a^m} = a^{n-m},\quad (a^n)^m = a^{nm}.$$
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Further generalizations of the idea of a power include: zero powers: $a^0 = 1 $,
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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
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[[De Moivre formula|de Moivre formula]]) and powers with a complex base and complex exponent (by definition: $z^u = e^{u\mathrm{Ln} z}$) are also studied.
  
 
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See also [[Exponential function]] and [[Exponential function, real]].
 
 
====Comments====
 
Besides the three properties listed above there is a fourth:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074190/p07419015.png" /></td> </tr></table>
 
 
 
Together these four operations are called the laws of exponents.
 

Latest revision as of 21:50, 31 December 2015

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.

See also Exponential function and Exponential function, real.

How to Cite This Entry:
Power. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power&oldid=15779
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article