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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.
In linear algebra, reduction refers to applying simple rules to a series of equations or matrices to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively.
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 ...
Notation for the (principal) square root of x. For example, √ 25 = 5, since 25 = 5 ⋅ 5, or 5 2 (5 squared). In mathematics, a square root of a number x is a number y such that =; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. [1]
In chemistry, a radical, also known as a free radical, is an atom, molecule, or ion that has at least one unpaired valence electron. [1] [2] With some exceptions, these unpaired electrons make radicals highly chemically reactive. Many radicals spontaneously dimerize. Most organic radicals have short lifetimes.
Mechanistic studies of reductions employing metal amalgams as the reducing agent suggest that upon electron transfer to the sulfone, fragmentation to a sulfinate anion and the more stable organic radical occurs. Immediate reduction of the radical and protonation then occur to afford the sulfur-free product derived from the more stable radical ...
Initial electron transfer and loss of halide generate an organic radical, which may combine with a second molecule of samarium(II) iodide to form an organosamarium species. Protonation of this species then yields the reduced product. Alternatively, the intermediate organic radical may abstract a hydrogen atom from the solvent S–H.
Conversely, the sodium is oxidized or is the electron donor, and thus induces a reduction in the other species and is considered the reducing agent. Which of the involved reactants would be a reducing or oxidizing agent can be predicted from the electronegativity of their elements. Elements with low electronegativities, such as most metals ...