# existence of $n$th root

###### Theorem.

If $a\mathrm{\in}\mathrm{R}$ with $a\mathrm{>}\mathrm{0}$ and $n$ is a positive integer, then there exists a unique positive real number $u$ such that ${u}^{n}\mathrm{=}a$.

###### Proof.

The statement is clearly true for $n=1$ (let $u=a$). Thus, it will be assumed that $n>1$.

Define $p:\mathbb{R}\to \mathbb{R}$ by $p(x)={x}^{n}-a$. Note that a positive real root of $p(x)$ corresponds to a positive real number $u$ such that ${u}^{n}=a$.

If $a=1$, then $p(1)={1}^{n}-1=0$, in which case the existence of $u$ has been established.

Note that $p(x)$ is a polynomial function and thus is continuous^{}. If $$, then $p(1)={1}^{n}-a>1-1=0$. If $a>1$, then $p(a)={a}^{n}-a=a({a}^{n-1}-1)>0$. Note also that $$. Thus, if $a\ne 1$, then the intermediate value theorem can be applied to yield the existence of $u$.

For uniqueness, note that the function $p(x)$ is strictly increasing on the interval $(0,\mathrm{\infty})$. It follows that $u$ as described in the statement of the theorem exists uniquely. ∎

Title | existence of $n$th root |
---|---|

Canonical name | ExistenceOfNthRoot |

Date of creation | 2013-03-22 15:52:15 |

Last modified on | 2013-03-22 15:52:15 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 21 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 26C10 |

Classification | msc 26A06 |

Classification | msc 12D99 |

Related topic | ExistenceOfNthRoot |