# 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 $\sqrt[5]{{x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}+5x+1}=x+1$ because ${(x+1)}^{5}={({x}^{2}+2x+1)}^{2}(x+1)={x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}+5x+1$. (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 ${z}_{1},{z}_{2},\mathrm{\dots},{z}_{n}\in \u2102$ 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 $\theta =\mathrm{arctan}\frac{y}{x}$ 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 |