Ads
related to: power rule with rational exponents
Search results
Results from the WOW.Com Content Network
Solving for , = = = = = Thus, the power rule applies for rational exponents of the form /, where is a nonzero natural number. This can be generalized to rational exponents of the form p / q {\displaystyle p/q} by applying the power rule for integer exponents using the chain rule, as shown in the next step.
For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below).
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the ...
The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e 0 = 1 , and e x is invertible with inverse e − x for any x in B .
It is not known whether n q is rational for any positive integer n and positive non-integer rational q. [20] For example, it is not known whether the positive root of the equation 4 x = 2 is a rational number. [citation needed] It is not known whether e π or π e (defined using Kneser's extension) are rationals or not.
The most general power rule is the functional power rule: for any functions and , ′ = () ′ = (′ + ′ ), wherever both sides are well defined. Special cases: If f ( x ) = x a {\textstyle f(x)=x^{a}} , then f ′ ( x ) = a x a − 1 {\textstyle f'(x)=ax^{a-1}} when a {\textstyle a} is any nonzero real number and x {\textstyle x} is ...
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
Ads
related to: power rule with rational exponents