# exponent

**exponent,**in mathematics, a number, letter, or algebraic expression written above and to the right of another number, letter, or expression called the base. In the expressions

*x*

^{2}and

*x*

^{ n }, the number 2 and the letter

*n*are the exponents respectively of the base

*x.*The exponent indicates the power to which the base is to be raised. When exponents were first introduced, only positive whole numbers were used, and the exponent indicated how many times the base was to be taken as a factor; e.g., 2

^{5}=32, or 2·2·2·2·2=32. In advanced algebra, fractions, zero, and negative numbers are also used as exponents. Particular meanings have been assigned to these types of exponents so that they obey the same algebraic rules as does the simpler type of exponent. A fractional exponent such as 1⁄4 or 1/

*n*indicates the fourth or

*n*th root , respectively, of the base. Any nonzero quantity raised to the zero power equals one; e.g.,

*x*

^{0}=5

^{0}=(

*a*

^{2}+

*b*

^{2})

^{0}=1. A negative exponent indicates the reciprocal of the quantity; e.g.,

*x*

^{−2}means 1/

*x*

^{2}. When quantities of the same base are multiplied together, their exponents are added; e.g.,

*x*

^{2}·

*x*

^{3}=

*x*

^{5}. Note that the base must be the same. When a quantity already containing an exponent is raised to a power, the exponents are multiplied; e.g., (

*x*

^{2})

^{3}=

*x*

^{6}.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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