section_path: Abstract
section_name: Abstract
paragraph_text: In mathematics, an n th root of a number x is a number r (the root) which, when raised to the power of the positive integer n, 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 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 n th root is a root extraction.
  
  For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9.
  
  The n th root of x is written as x n {\displaystyle {\sqrt{x}}} using the radical symbol or radix x {\displaystyle {\sqrt {\phantom {x}}}}. The square root is usually written without the n as just x {\displaystyle {\sqrt {x}}}. Taking the n th root of a number is the inverse operation of exponentiation, and can be written as a fractional exponent:
  
  x n = x 1 / n. {\displaystyle {\sqrt{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{x}}} denotes the positive real n th root. 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.
  
  In general, any non-zero complex number has n distinct complex-valued n th roots, equally distributed around a complex circle of constant absolute value. (The n th root of 0 is zero with multiplicity n, and this circle degenerates to a point.) Extracting the n th roots of a complex number x can thus be taken to be a multivalued function. By convention the principal value of this function, called the principal root and denoted x n {\displaystyle {\sqrt{x}}}, is taken to be the n th root with the greatest real part and in the special case when x is a negative real number, the one with a positive imaginary part. The principal root of a positive real number is thus also a positive real number. As a function, the principal root is continuous in the whole complex plane, except along the negative real axis.
  
  An unresolved root, especially one using the radical symbol, is sometimes 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.
  
  Roots are used for determining the radius of convergence of a power series with the root test. The n th roots of 1 are called roots of unity and play a fundamental role in various areas of mathematics, such as number theory, theory of equations, and Fourier transform.