nth root

The phrase “the n-th root of a number” is a somewhat misleading concept that requires a fair amount of thought to make rigorous.

For n a positive integer, we define an n-th root of a number x to be a number y such that yn=x. The number n is said to be the index of the root. Note that the term “number” here is ambiguous, as the discussion can apply in a varietyMathworldPlanetmath of contexts (groups, rings, monoids, etc.) The purpose of this entry is specifically to deal with n-th roots of real and complex numbersMathworldPlanetmathPlanetmath.

In an effort to give meaning to the term the n-th root of a real number x, we define it to be the unique real number that y is an nth root of x and such that sign(x)=sign(y), if such a number exists. We denote this number by xn, or by x1n if x is positive. This specific nth root is also called the principal nth root.

Example: 814=3 because 34=3×3×3×3=81, and 3 is the unique positive real number with this property.

Example: If x+1 is a positive real number, then we can write x5+5x4+10x3+10x2+5x+15=x+1 because (x+1)5=(x2+2x+1)2(x+1)=x5+5x4+10x3+10x2+5x+1. (See the Binomial Theorem and .)

The nth root operationMathworldPlanetmath is distributive for multiplicationPlanetmathPlanetmath and division, but not for addition and subtractionPlanetmathPlanetmath. That is, x×yn=xn×yn, and xyn=xnyn. However, except in special cases, x+ynxn+yn and x-ynxn-yn.

Example: 816254=35 because (35)4=3454=81625.

Note that when we restrict our attention to real numbers, expressions like -3 are undefined. Thus, for a more full definition of nth roots, we will have to incorporate the notion of complex numbers: The nth roots of a complex number t=x+yi are all the complex numbers z1,z2,,zn that satisfy the condition zkn=t. Applying the fundamental theorem of algebra (complex version) to the function xn-t tells us that n such complex numbers always exist (counting multiplicityMathworldPlanetmath).

One of the more popular methods of finding these roots is through trigonometryMathworldPlanetmath and the geometryMathworldPlanetmathPlanetmath of complex numbers. For a complex number z=x+iy, recall that we can put z in polar form: z=(r,θ), where r=x2+y22, and θ=π2 if x=0, and θ=arctanyx if x0. (See the Pythagorean TheoremMathworldPlanetmathPlanetmath.) For the specific procedures involved, see calculating the nth roots of a complex number.

Title nth root
Canonical name NthRoot
Date of creation 2013-03-22 11:57:27
Last modified on 2013-03-22 11:57:27
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 25
Author mathcam (2727)
Entry type Definition
Classification msc 30-00
Classification msc 12D99
Synonym complex root
Synonym principal root
Related topic SquareRoot
Related topic CubeRoot
Related topic RealNumber
Related topic RationalNumber
Related topic Complex
Related topic IrrationalNumber
Related topic EvenEvenOddRule
Related topic ExtensionOfValuationFromCompleteBaseField
Related topic Radical5
Related topic Radical6
Related topic ExampleOfConvergingIncreasingSequence
Related topic NthRootByNewtonsMethod
Defines index