Product Rule for Radicals

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:

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

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.

The computation of an nth root is a root extraction.

For example:

• 3 is a square root of 9, since 3² = 9.
• −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 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 (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 with ${\displaystyle {\sqrt {x}}}$ denoting the positive square root of x if x is positive, and ${\displaystyle {\sqrt[{n}]{x}}}$ denoting the real nth root, if n is odd, and the positive square root if n is even and x is nonnegative. In the other cases, the symbol is not commonly used as being ambiguous. In the expression ${\displaystyle {\sqrt[{n}]{x}}}$, the integer n is called the index, ${\displaystyle {\sqrt {{~^{~}}^{~}\!\!}}}$ is the radical sign or radix, and x is called the radicand.

When complex nth roots are considered, it is often useful to choose one of the roots as a principal value. The common choice is the one that makes the nth root a continuous function that is real and positive for x real and positive. More precisely, the principal nth root of x is 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.

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 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:

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

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.