Nth root

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In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as ${\displaystyle {\sqrt[{n}]{x}}}$, where x is the radicand and n is the index (also sometimes called the degree). This is pronounced as "the nth root of x". The definition then of 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:

$${\displaystyle r^{n}=x.}$$

A root of degree 2 is called a square root (usually written without the n as just ${\displaystyle {\sqrt {x}}}$) and a root of degree 3, a cube root (written ${\displaystyle {\sqrt[{3}]{x}}}$). 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.

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.

Any non-zero number considered as a complex number has n different complex nth roots, including the real ones (at most two). The nth root of 0 is zero for all positive integers n, since 0ⁿ = 0. In particular, if n is even and x is a positive real number, one of its nth roots is real and positive, one is negative, and the others (when n > 2) are non-real complex numbers; if n is even and x is a negative real number, none of the nth roots are real. If n is odd and x is real, one nth 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 nth roots are real.

Roots of real numbers are usually written using the radical symbol or radix ${\displaystyle {\sqrt {{~^{~}}^{~}\!\!}}}$, with ${\displaystyle {\sqrt {x}}}$ denoting the positive square root of x if x is positive; for higher roots, ${\displaystyle {\sqrt[{n}]{x}}}$ denotes the real nth root if n is odd, and the positive nth root if n is even and x is positive. In the other cases, the symbol is not commonly used as being ambiguous.

When complex nth roots are considered, it is often useful to choose one of the roots, called principal root, as a principal value. The common choice is to choose the principal nth root of x as the nth root with the greatest real part, and when there are two (for x real and negative), the one with a positive imaginary part. This makes the nth root a function that is real and positive for x real and positive, and is continuous in the whole complex plane, except for values of x that are real and negative.

A difficulty with this choice is that, for a negative real number and an odd index, the principal nth root is not the real one. For example, ${\displaystyle -8}$ has three cube roots, ${\displaystyle -2}$, ${\displaystyle 1+i{\sqrt {3}}}$ and ${\displaystyle 1-i{\sqrt {3}}.}$ The real cube root is ${\displaystyle -2}$ and the principal cube root is ${\displaystyle 1+i{\sqrt {3}}.}$

An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd^[1] or a radical.^[2] 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.

The positive root of a number is the inverse operation of exponentiation with positive integer exponents.^[3] Roots can also be defined as special cases of exponentiation, where the exponent is a fraction:

$${\displaystyle {\sqrt[{n}]{x}}=x^{1/n}.}$$

Arithmetic operations v t e

Addition (+) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{sum}}}$ Subtraction (−) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{difference}}}$ Multiplication (×) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{product}}}$ Division (÷) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}$ Exponentiation (^) ${\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,}$ ${\displaystyle \scriptstyle {\text{power}}}$ nth root (√) ${\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}$ ${\displaystyle \scriptstyle {\text{root}}}$ Logarithm (log) ${\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}$ ${\displaystyle \scriptstyle {\text{logarithm}}}$

v t e

Roots are used for determining the radius of convergence of a power series with the root test. The nth 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.