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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 ( times), where is the base, the exponent and the power. The basic properties of powers are:

Further generalizations of the idea of a power include: zero powers: (when ); negative powers: ; fractional powers: ; and a power with an irrational exponent: , where is an arbitrary sequence of rational numbers tending to . Powers with a complex base (see de Moivre formula) and powers with a complex base and complex exponent (by definition: ) are also studied.


Besides the three properties listed above there is a fourth:

Together these four operations are called the laws of exponents.

How to Cite This Entry:
Power. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article