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Exponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b: =. [1] Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (). [26]
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 ...
When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base. [2] Thus 3 + 5 2 = 28 and 3 × 5 2 = 75. These conventions exist to avoid notational ambiguity while allowing notation to remain brief. [4]
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To find the number of negative roots, change the signs of the coefficients of the terms with odd exponents, i.e., apply Descartes' rule of signs to the polynomial = + + This polynomial has two sign changes, as the sequence of signs is (−, +, +, −) , meaning that this second polynomial has two or zero positive roots; thus the original ...
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.Two definitions of a monomial may be encountered: A monomial, also called a power product or primitive monomial, [1] is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. [2]
There is no standard notation for tetration, though Knuth's up arrow notation and the left-exponent are common. Under the definition as repeated exponentiation, n a {\displaystyle {^{n}a}} means a a ⋅ ⋅ a {\displaystyle {a^{a^{\cdot ^{\cdot ^{a}}}}}} , where n copies of a are iterated via exponentiation, right-to-left, i.e. the application ...
If we allow some real coefficients a n, to get the form ()it is the same as allowing exponents that are complex numbers.Both forms are certainly useful in applications. A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work of Hermann Weyl in diophantine approximation.