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The nested square roots of 2 are a special case of the wide class of infinitely nested radicals. There are many known results that bind them to sines and cosines . For example, it has been shown that nested square roots of 2 as [ 7 ] R ( b k , … , b 1 ) = b k 2 2 + b k − 1 2 + b k − 2 2 + ⋯ + b 2 2 + x {\displaystyle R(b_{k},\ldots ,b ...
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
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
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 ...
As discussed in § Constructibility, only certain angles that are rational multiples of radians have trigonometric values that can be expressed with square roots. The angle 1°, being π / 180 = π / ( 2 2 ⋅ 3 2 ⋅ 5 ) {\displaystyle \pi /180=\pi /(2^{2}\cdot 3^{2}\cdot 5)} radians, has a repeated factor of 3 in the denominator and therefore ...
Radical extensions occur naturally when solving polynomial equations in radicals.In fact a solution in radicals is the expression of the solution as an element of a radical series: a polynomial f over a field K is said to be solvable by radicals if there is a splitting field of f over K contained in a radical extension of K.
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