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In mathematics, an nth 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. A root of degree 2 is called a square root and a root of degree 3, a cube root.
Then let =, where the symbols and are interpreted as any square root and any cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign " ± " before the square root is either " + " or " – "; the choice is almost arbitrary, and changing it amounts to choosing a different square root.
Cube root. In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other ...
Nested radical. 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. Examples include. which arises in discussing the regular pentagon, and more complicated ones such as.
Rationalisation of a monomial square root and cube root For the fundamental technique, the numerator and denominator must be multiplied by the same factor. Example 1: To rationalise this kind of expression, bring in the factor : The square root disappears from the denominator, because by definition of a square root:
The square root function maps rational numbers into algebraic numbers, the latter being a superset of the rational numbers). For all real numbers x, (see absolute value). For all nonnegative real numbers x and y, and. The square root function is continuous for all nonnegative x, and differentiable for all positive x.
Polynomial transformation. In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.
The angle 1°, being radians, has a repeated factor of 3 in the denominator and therefore cannot be expressed using only square roots. A related question is whether it can be expressed using cube roots. The following two approaches can be used, but both result in an expression that involves the cube root of a complex number.
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