**Exponentiation** is a mathematical operation, written as ** bn**, involving two numbers, the

**base**

*b*and the

**exponent**(or

**index**or

**power**)

*n*. When

*n*is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of

*factors, each of which is equal to*

**n***(the product itself can also be called*

**b****power**):

just as multiplication by a positive integer corresponds to repeated addition:

The exponent is usually shown as a superscript to the right of the base. The exponentiation *b**n* can be read as: * b raised to the n-th power*,

*, or*

**b**raised to the power of**n***, most briefly as*

**b**raised by the exponent of**n***. Some exponents have their own pronunciation: for example,*

**b**to the**n***b*2 is usually read as

*and*

**b**squared*b*3 as

*.*

**b**cubedThe power *b**n* can be defined also when *n* is a negative integer, for nonzero *b*. No natural extension to all real *b* and *n* exists, but when the base *b* is a positive real number, *b**n* can be defined for all real and even complex exponents *n* via the exponential function *e**z*. Trigonometric functions can be expressed in terms of complex exponentiation.

Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations.

Exponentiation is used pervasively in many other fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

Read more about Exponentiation: Background and Terminology, Rational Exponents, Real Exponents, Powers of Complex Numbers, Limits of Powers, Efficient Computation of Integer Powers, Exponential Notation For Function Names, Repeated Exponentiation, In Programming Languages, History of The Notation

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