## You are here

Homeexistence of $n$th root

## Primary tabs

# existence of $n$th root

###### Theorem.

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

###### Proof.

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

Define $p\colon\mathbb{R}\to\mathbb{R}$ 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 $a<1$, 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 $p(0)=0^{n}-a=-a<0$. Thus, if $a\neq 1$, 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 $(0,\infty)$. It follows that $u$ as described in the statement of the theorem exists uniquely. ∎

## Mathematics Subject Classification

26C10*no label found*26A06

*no label found*12D99

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith