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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.
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
By systematically simplifying radicals, large swaths of the character set are altered. Some simplifications were based on popular cursive forms that embody graphic or phonetic simplifications of the traditional forms. In addition, variant characters with identical pronunciation and meaning were reduced to a single standardized character ...
Simplifying radical expressions involving nested radicals can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced Galois theory . Moreover, when complete denesting is impossible, there is no general canonical form such that the equality of two numbers can be tested by simply looking at ...
Radical expression involving roots, also known as an nth root; Radical symbol (√), used to indicate the square root and other roots; Radical of an algebraic group, a concept in algebraic group theory; Radical of an ideal, an important concept in abstract algebra; Radical of a ring, an ideal of "bad" elements of a ring
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
The 辶 radical was previously printed with two dots (as in the hyōgaiji 逞) but was written with one (as in 道), so the written form with one dot became standard. The upper 丷 portion of the characters 半, 尊, and 平 was previously printed as 八 and written 丷 (as in the aforementioned examples), but the old printed form is still seen ...