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Pic.1: Schematic triangle diagram of application of a rewrite rule at position in a term, with matching substitution Pic.2: Rule lhs term () matching in term ((+) (+)) () A term rewriting system ( TRS ) is a rewriting system whose objects are terms , which are expressions with nested sub-expressions.
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 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}
In elementary algebra, root rationalisation (or rationalization) is a process by which radicals in the denominator of an algebraic fraction are eliminated.. If the denominator is a monomial in some radical, say , with k < n, rationalisation consists of multiplying the numerator and the denominator by , and replacing by x (this is allowed, as, by definition, a n th root of x is a number that ...
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 ...
Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d.
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
If the latter (→) is moreover well-founded, it is called a reduction ordering, [2] or a reduction preorder. Given a binary relation R, its rewrite closure is the smallest rewrite relation containing R. [3] A transitive and reflexive rewrite relation that contains the subterm ordering is called a simplification ordering. [4]