Exponent.

Exponent. Exponent, term used in mathematics to indicate the number of times that a quantity is to be multiplied by itself. An exponent is usually written as a small number or letter at the upper right of the base expression, as in x2, which is read x squared and means x times x. Terms containing exponents are also often referred to as powers of their bases--x2, therefore, can be read x to the second power. Similarly, x3 is read x cubed or x to the third power and means x times x times x. In computation, exponents obey certain rules called the laws of exponents: (1) xm · xn = xm+n For example, 22 · 23 = 22+3 = 25. (2) xm · ym = (xy)m For example, 23 · 33 = (2 · 3)3 = 63. (3) (xm)n = xm·n For example, (22)3 = 22·3 = 26. Exponents may be positive or negative integers or zero, rational numbers, irrational numbers, or complex numbers. A negative exponent indicates that the reciprocal of the expression is to be taken--that is, x-n = 1/xn. Any base with the exponent 0 is equal to one. Thus, x0 = 1 for all values of x except 0 (00 is undefined). Any base with the exponent 1 is equal to itself: x1 = x. Taking the root of a number is the opposite of raising a number to a power. An n th root of x, written n?x, is a number whose n th power is x. For example, 3?8 = 2 since 23 = 8. A fractional exponent of the form x1/n indicates that the n th root of a number is to be taken. For example, 2y stands for the square root of two, 2?the fourth root of two, and so on. A fractional exponent of the form xm/n indicates taking the n th root of x raised to the mth power: n?xm. For example, 22/3 = 3?22. Since n?xm = (n?x)m, either operation may be performed first. Mathematicians used a system of exponents as early as the 14th century, but the modern concept was not in general use until the work of the French mathematician René Descartes in the 17th century. Contributed By: J. Lennart Berggren Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.