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The first negative powers of 2 have special names: is a half; is a quarter. Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2 n members. Integer powers of 2 are important in computer science.
By comparison, powers of two with negative exponents are fractions: for positive integer n, 2-n is one half multiplied by itself n times. Thus the first few negative powers of 2 are 1 / 2 , 1 / 4 , 1 / 8 , 1 / 16 , etc. Sometimes these are called inverse powers of two because each is the multiplicative inverse of a ...
This necessarily leads to the same result. Note that because () does not have a conventional definition when is not a rational number, irrational power functions are not well defined for negative bases. In addition, as rational powers of −1 with even denominators (in lowest terms) are not real numbers, these expressions are only real valued ...
The term hyperpower [4] is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence. When considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration.
Power(x, −n) = Power(x −1, n), Power(x, −n) = (Power(x, n)) −1. The approach also works in non-commutative semigroups and is often used to compute powers of matrices. More generally, the approach works with positive integer exponents in every magma for which the binary operation is power associative.
Negative powers are not permitted in an ordinary power series; for instance, + + + + is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as x 1 2 {\textstyle x^{\frac {1}{2}}} are not permitted; fractional powers arise in Puiseux series .
The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, [2] cloud sizes, [3] the foraging pattern of various species, [4] the sizes of activity patterns of neuronal populations, [5] the frequencies of words in most languages ...
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 ...
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