# 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 $\operatorname{sign}(x)=\operatorname{sign}(y)$, 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, $\sqrt[n]{x+y}\not=\sqrt[n]{x}+\sqrt[n]{y}$ and $\sqrt[n]{x-y}\not=\sqrt[n]{x}-\sqrt[n]{y}$.

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 $z_{1},z_{2},\ldots,z_{n}\in\mathbb{C}$ 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 $x\not=0$. (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