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In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression ...
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 ...
For degrees three and four, there are closed-form solutions in terms of radicals, which are generally not convenient for numerical evaluation, as being too complicated and involving the computation of several n th roots whose computation is not easier than the direct computation of the roots of the polynomial (for example the expression of the ...
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 ...
Moreover, there exist more informative radical expressions for n th roots of unity with the additional property [12] that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive n th root of unity. This was already shown by Gauss in 1797. [13]
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 radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative x , the principal square root can also be written in exponent notation, as x 1 / 2 {\displaystyle x^{1/2}} .
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).