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In mathematics, a sum of radicals is defined as a finite linear combination of n th roots: =, where , are natural numbers and , are real numbers.. A particular special case arising in computational complexity theory is the square-root sum problem, asking whether it is possible to determine the sign of a sum of square roots, with integer coefficients, in polynomial time.
SRS can be solved in polynomial time in the Real RAM model. [3] However, its run-time complexity in the Turing machine model is open, as of 1997. [1] The main difficulty is that, in order to solve the problem, the square-roots should be computed to a high accuracy, which may require a large number of bits.
Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to the power of 1 2 {\displaystyle {\tfrac {1}{2}}} and the cube root of a number is the same as raising the number to the power of 1 3 {\displaystyle {\tfrac {1}{3}}} .
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R.Then, the quotients / belong to the field of fractions of R (and possibly are in R itself if happens to be invertible in R) and the roots are taken in an algebraically closed extension.
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 fact, the n th roots of unity being the roots of the polynomial X n – 1, their sum is the coefficient of degree n – 1, which is either 1 or 0 according whether n = 1 or n > 1. Alternatively, for n = 1 there is nothing to prove, and for n > 1 there exists a root z ≠ 1 – since the set S of all the n th roots of unity is a group , z S ...
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In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.