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 $y^n=x$. 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 variety of contexts (groups, rings, monoids, etc.) The purpose of this entry is specifically to deal with $n$-th roots of real and complex numbers.

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 $n$th root of $x$ and such that $\mathrm{sign}(x)=\mathrm{sign}(y)$, if such a number exists. We denote this number by $\sqrt[n]{x}$, or by $x^{\frac{1}{n}}$ if $x$ is positive. This specific $n$th root is also called the principal $n$th root.

Example: $\sqrt[4]{81}=3$ because $3^4=3\times 3\times 3\times 3=81$, and $3$ is the unique positive real number with this property.

Example: If $x+1$ is a positive real number, then we can write $\sqrt[5]{x^5+5x^4+10x^3+10x^2+5x+1}=x+1$ because ${(x+1)}^5={(x^2+2x+1)}^2(x+1)=x^5+5x^4+10x^3+10x^2+5x+1$. (See the Binomial Theorem and .)

The nth root operation is distributive for multiplication and division, but not for addition and subtraction. That is, $\sqrt[n]{x\times y}=\sqrt[n]{x}\times \sqrt[n]{y}$, and $\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}$. However, except in special cases, $\sqrt[n]{x+y}\ne \sqrt[n]{x}+\sqrt[n]{y}$ and $\sqrt[n]{x-y}\ne \sqrt[n]{x}-\sqrt[n]{y}$.

Example: $\sqrt[4]{\frac{81}{625}}=\frac{3}{5}$ because ${\left(\frac{3}{5}\right)}^4=\frac{3^4}{5^4}=\frac{81}{625}$.

Note that when we restrict our attention to real numbers, expressions like $\sqrt{-3}$ are undefined. Thus, for a more full definition of $n$th 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 $z_1,z_2,\mathrm{\dots },z_n\in ℂ$ that satisfy the condition $z_k^n=t$. Applying the fundamental theorem of algebra (complex version) to the function $x^n-t$ tells us that $n$ such complex numbers always exist (counting multiplicity).

One of the more popular methods of finding these roots is through trigonometry and the geometry of complex numbers. For a complex number $z=x+iy$, recall that we can put $z$ in polar form: $z=(r,\theta )$, where $r=\sqrt[2]{x^2+y^2}$, and $\theta =\frac{\pi }{2}$ if $x=0$, and $\theta =\arctan \frac{y}{x}$ if $x\ne 0$. (See the Pythagorean Theorem.) 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