radical (mathematics)

See radical for other uses of the term

In mathematics, the nth root or radical of the non-negative real number a, written as

\sqrt n {a}

, is the unique non-negative real number b such that bⁿ = a. See square root for the case where n = 2.

Fundamental operations

Operations with radicals are given by the following formulas:

\sqrtn{ab} = \sqrtn{a} \sqrtn{b},

\sqrt n {\frac{a}{b}} = \frac{\sqrt n {a}}{\sqrt n {b}},

\sqrtn{a^m} = \left(\sqrtn{a}\right)^m = a^{\frac{m}{n}}, where a and b are positive. For every non-zero complex number a, there are n different complex numbers b such that bⁿ = a, so the symbol

\sqrt n {a}

cannot be used unambiguously. The nth roots of unity are of particular importance.

Working with surds

Often in calculations, it pays to leave the square or other roots of numbers unresolved. One can then manipulate these unresolved expressions into simpler forms or arrange them to cancel each other. Notationally, the √ symbol depicts surds, with an upper line above the expression (called the vinculum) enclosed in the root. A cube root takes the form:

\sqrt 3 {a}

, which corresponds to

a^\frac{1}{3}

, when expressed using indices.

Square roots, cube roots and so on, can all remain in surd form. Basic techniques for working with surds arise from identities. Typical examples include:

\sqrt{a^2 b} = a \sqrt{b}

\sqrt n {a^m b} = a^{\frac{m}{n}}\sqrt n {b}

\sqrt{a} \sqrt{b} = \sqrt{ab}

(\sqrt{a}+\sqrt{b})^{-1} = \frac{1}{(\sqrt{a}+\sqrt{b})} = \frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})} = \frac{\sqrt{a}- \sqrt{b}} {a - b}

The last of these can serve to rationalise the denominator of an expression, moving surds from the denominator to the numerator. It follows from the identity

(\sqrt{a}+\sqrt{b})(\sqrt{a}- \sqrt{b}) = a - b

which exemplifies a case of the difference of two squares. A cube root variant exists, as do more general formulae based on summing a finite geometric progression.

Radical (Mathematics)

Fundamental operations

Working with surds

See also