Nth roots

In mathematics, an n-th root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x:

where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc.

For example:

Any non-zero number, considered as complex number, has n different "complex roots of degree n" (n-th roots), including those with zero imaginary part, i.e. any real roots. The root of 0 is zero for all degrees n, since 0ⁿ = 0. In particular, if n is even and x is a positive real number, one of its n-th roots is positive, one is negative, and the rest (when n > 2) are complex but not real; if n is even and x is a negative real, none of the n-th roots is real. If n is odd and x is real, one n-th root is real and has the same sign as x, while the other (n − 1) roots are not real. Finally, if x is not real, then none of its n-th roots is real.

Roots are usually written using the radical symbol or radix with ${\displaystyle {\sqrt {x}}}$ denoting the principal square root of ${\displaystyle x}$, ${\displaystyle {\sqrt[{3}]{x}}}$ denoting the principal cube root, ${\displaystyle {\sqrt[{4}]{x}}}$ denoting the principal fourth root, and so on. In the expression ${\displaystyle {\sqrt[{n}]{x}}}$, n is called the index, ${\displaystyle {\sqrt {\;}}}$ is the radical sign or radix, and ${\displaystyle x}$ is called the radicand. Since the radical symbol denotes a function, it is defined to return only one result for a given argument ${\displaystyle x}$, which is called the principal n-th root of ${\displaystyle x}$. Conventionally, a real root, preferably non-negative, if there is one, is designated as the principal n-th root.

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