In mathematics, an nth 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: \n* 2 is a square root of 4, since 22 = 4. \n* −2 is also a square root of 4, since (−2)2 = 4. A real number or complex number has n complex roots of degree n. While the roots of 0 are not distinct (all equaling 0), the n nth roots of any other real or complex number are all distinct. If n is even and x is real and positive, one of its nth roots is positive, one is negative, and the rest are either non-existent (in the case when n = 2) or complex but not real; if n is even and x is real and negative, none of the nth roots is real. If n is odd and x is real, one nth root is real and has the same sign as x , while the other roots are not real. Finally, if x is not real, then none of its nth roots is real. Roots are usually written using the radical symbol or radix or , with or denoting the square root, denoting the cube root, denoting the fourth root, and so on. In the expression , n is called the index, is the radical sign or radix, and x is called the radicand. Since the radical symbol denotes a function, when a number is presented under the radical symbol it must return only one result, so a non-negative real root, called the principal nth root, is preferred rather than others; if the only real root is negative, as for the cube root of –8, again the real root is considered the principal root. An unresolved root, especially one using the radical symbol, is often referred to as a surd or a radical. 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 expression. In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction: Roots are particularly important in the theory of infinite series; the root test determines the radius of convergence of a power series. Nth roots can also be defined for complex numbers, and the complex roots of 1 (the roots of unity) play an important role in higher mathematics. Galois theory can be used to determine which algebraic numbers can be expressed using roots, and to prove the Abel-Ruffini theorem, which states that a general polynomial equation of degree five or higher cannot be solved using roots alone; this result is also known as "the insolubility of the quintic".

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