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 $p:ℝ\to ℝ$ 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