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The result is an equation with no fractions. The simplified equation is not entirely equivalent to the original. For when we substitute y = 0 and z = 0 in the last equation, both sides simplify to 0, so we get 0 = 0, a mathematical truth. But the same substitution applied to the original equation results in x/6 + 0/0 = 1, which is ...
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.
Factorization is one of the most important methods for expression manipulation for several reasons. If one can put an equation in a factored form E⋅F = 0, then the problem of solving the equation splits into two independent (and generally easier) problems E = 0 and F = 0. When an expression can be factored, the factors are often much simpler ...
In mathematics, an n th root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields x: = ⏟ =.. The integer n is called the index or degree, and the number x of which the root is taken is the radicand.
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.
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
Note that even simple equations like = are solved using cross-multiplication, since the missing b term is implicitly equal to 1: =. Any equation containing fractions or rational expressions can be simplified by multiplying both sides by the least common denominator.
For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. [2] As another example, if a × b = a × c , then the multiplicative term a can be canceled out if a ≠0, resulting in the equivalent expression b = c ; this is equivalent to dividing through by a .