Saturday, March 12, 2011

Exponentiation



Exponentiation is a mathematical operation, written as an, involving two numbers, the base 'a' and the exponent 'n'.

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

a^n = \underbrace{a \times \cdots \times a}_n,

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

a \times n = \underbrace{a + \cdots + a}_n.

The exponent is usually shown as a superscript to the right of the base.

The power an can be defined also when n is a negative integer, for nonzero a. No natural extension to all real a and n exists, but when the base a is a positive real number, an can be defined for all real and even complex exponents n via the exponential function ez.

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.

Graphs of y=ax for various bases a: base 10 (green), base e (red), base 2 (blue), and base ½ (cyan). Each curve passes through the point (0,1) because any nonzero number raised to the power 0 is 1. At x=1, the y-value equals the base because any number raised to the power 1 is itself.

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