Nth root

In mathematics, an nth root of a number x is the number r which, when n copies are multiplied together, yields x: r n = r × r × ⋯ × r ⏟ n factors = x . {\displaystyle r^{n}=\underbrace {r\times r\times \dotsb \times r} _{n{\text{ factors}}}=x.} The positive 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. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an nth root is a root extraction. The nth root of x is written as x n {\displaystyle {\sqrt[{n}]{x}}} using the radical symbol x {\displaystyle {\sqrt {\phantom {x}}}} . The square root is usually written as ⁠ x {\displaystyle {\sqrt {x}}} ⁠, with the degree omitted. Taking the nth root of a number, for fixed ⁠ n {\displaystyle n} ⁠, is the inverse of raising a number to the nth power, and can be written as a fractional exponent: x n = x 1 / n . {\displaystyle {\sqrt[{n}]{x}}=x^{1/n}.} For a positive real number x, x {\displaystyle {\sqrt {x}}} denotes the positive square root of x and x n {\displaystyle {\sqrt[{n}]{x}}} denotes the positive real nth root. For example, 3 is a square root of 9, since 32 = 9, and −3 is also a square root of 9, since (−3)2 = 9. A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, ⁠ + i x {\displaystyle +i{\sqrt {x}}} ⁠ and ⁠ − i x {\displaystyle -i{\sqrt {x}}} ⁠, where i is the imaginary unit....