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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.). A well-known example is the quadratic formula
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.
Radical elimination can be viewed as the reverse of radical addition. In radical elimination, an unstable radical compound breaks down into a spin-paired molecule and a new radical compound. Shown below is an example of a radical elimination reaction, where a benzoyloxy radical breaks down into a phenyl radical and a carbon dioxide molecule. [7]
In the second step, the distributive law is used to simplify each of the two terms. Note that this process involves a total of three applications of the distributive property. In contrast to the FOIL method, the method using distributivity can be applied easily to products with more terms such as trinomials and higher.
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 ...
The final step is called termination (6,7), in which the radical recombines with another radical species. If the reaction is not terminated, but instead the radical group(s) go on to react further, the steps where new radicals are formed and then react are collectively known as propagation (4,5). This is because a new radical is created, able ...
The formed product, a carbon radical, can react with non-radical molecule to continue propagation or react with another radical to form a new stable molecule such as a longer carbon chain or an alkyl halide. [4] The example below of methane chlorination shows a multi-step reaction involving radicals.
Radicals can be formed by a number of different mechanisms, usually involving separate initiator molecules. Following its generation, the initiating radical adds (nonradical) monomer units, thereby growing the polymer chain. Radical polymerization is a key synthesis route for obtaining a wide variety of different polymers and materials ...
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