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This definition of exponentiation with negative exponents is the only one that allows extending the identity + = to negative exponents (consider the case =). The same definition applies to invertible elements in a multiplicative monoid , that is, an algebraic structure , with an associative multiplication and a multiplicative identity denoted 1 ...
For example, from the differential equation definition, e x e −x = 1 when x = 0 and its derivative using the product rule is e x e −x − e x e −x = 0 for all x, so e x e −x = 1 for all x. From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity.
Zero to the power of zero, denoted as 0 0, is a mathematical expression that can take different values depending on the context. In certain areas of mathematics, such as combinatorics and algebra , 0 0 is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents .
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of f.The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents.
For example, 2 tetrated to 4 (or the fourth tetration of 2) is = = = =. It is the next hyperoperation after exponentiation , but before pentation . The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration .
An algebraic number is a number that is the root of some non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. algebraic number field See number field. algebraic number theory Algebraic number theory analytic number theory Analytic number theory Artin
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