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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}
An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd [2] or a radical. [3] Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression , and if it contains no transcendental functions or transcendental numbers it is called an algebraic ...
There are 252 points on the surface of a cuboctahedron of radius five in the face-centered cubic lattice, [8] 252 ways of writing the number 4 as a sum of six squares of integers, [9] 252 ways of choosing four squares from a 4×4 chessboard up to reflections and rotations, [10] and 252 ways of placing three pieces on a Connect Four board. [11]
The radical symbol refers to the principal value of the square root function called the principal square root, which is the positive one. The two square roots of a negative number are both imaginary numbers , and the square root symbol refers to the principal square root, the one with a positive imaginary part.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.).
In algebra, the real radical of an ideal I in a polynomial ring with real coefficients is the largest ideal containing I with the same (real) vanishing locus. It plays a similar role in real algebraic geometry that the radical of an ideal plays in algebraic geometry over an algebraically closed field .
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.
The radical of an integer is its largest square-free factor, that is = with notation of the preceding section. An integer is square-free if and only if it is equal to its radical. Every positive integer n {\displaystyle n} can be represented in a unique way as the product of a powerful number (that is an integer such that is divisible by the ...