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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.
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by The expansion is given by ( a + b + c ) n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},}
Exponentiation is written as b n, where b is the base and n is the power; often said as "b to the power n ". [1] When n is a positive integer , exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [ 1 ] b n = b × b × ⋯ × b × b ⏟ n times . {\displaystyle b^{n}=\underbrace ...
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 ...
For example, on a simple calculator, typing 1 + 2 × 3 = yields 9, while a more sophisticated calculator will use a more standard priority, so typing 1 + 2 × 3 = yields 7. Calculators may associate exponents to the left or to the right. For example, the expression a^b^c is interpreted as a (b c) on the TI-92 and the TI-30XS MultiView in ...
Semi-log plot of solutions of + + = for integer , , and , and .Green bands denote values of proven not to have a solution.. In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum.
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Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to , and 25.12 is equal to . [20] Every rational number corresponds to a finite or a repeating decimal. [21] [c]