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Repeated application of the half-angle formulas leads to nested radicals, specifically nested square roots of 2 of the form . In general, the sine and cosine of most angles of the form β / 2 n {\displaystyle \beta /2^{n}} can be expressed using nested square roots of 2 in terms of β {\displaystyle \beta } .
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as 3 {\textstyle {\sqrt {3}}} or 3 1 / 2 {\displaystyle 3^{1/2}} . It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property.
Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equation x 5 − 5x 4 + 30x 3 − 50x 2 + 55x − 21 = 0, for which the only real solution is
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points.
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation in the form of square roots and fourth roots. [21]
Consider the ring of integers.. The radical of the ideal of integer multiples of is (the evens).; The radical of is .; The radical of is .; In general, the radical of is , where is the product of all distinct prime factors of , the largest square-free factor of (see Radical of an integer).
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.
One can prove [citation needed] that = is the largest possible number for which the stopping criterion | + | < ensures ⌊ + ⌋ = ⌊ ⌋ in the algorithm above.. In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than 1 should be used to protect against round-off errors.