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However, for any degree there are some polynomial equations that have algebraic solutions; for example, the equation = can be solved as =. The eight other solutions are nonreal complex numbers , which are also algebraic and have the form x = ± r 2 10 , {\displaystyle x=\pm r{\sqrt[{10}]{2}},} where r is a fifth root of unity , which can be ...
The positive integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction.
In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: r a d ( n ) = ∏ p ∣ n p prime p {\displaystyle \displaystyle \mathrm {rad} (n)=\prod _{\scriptstyle p\mid n \atop p{\text{ prime}}}p}
Repeated application of the half-angle formulas leads to nested radicals, specifically nested square roots of 2 of the form . In general, the sine and cosine of most angles of the form β / 2 n {\displaystyle \beta /2^{n}} can be expressed using nested square roots of 2 in terms of β {\displaystyle \beta } .
is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals. The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the quadratic formula , the cubic formula , and the quartic formula for degrees two, three, and ...
In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form , with q and p being coprime integers. The rational root test allows finding q and p by examining a finite number of cases (because q must be a divisor of a , and p must be a divisor of d ).
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.
Pages in category "Simplified Chinese radicals" ... Radical 80; Radical 113; Radical 99; Radical 112; Radical 109; Radical 102; Radical 122; Radical 108; Radical 100 ...