existence of nth root


If aR with a>0 and n is a positive integer, then there exists a unique positive real number u such that un=a.


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

Define p: by p(x)=xn-a. Note that a positive real root of p(x) corresponds to a positive real number u such that un=a.

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

Note that p(x) is a polynomial function and thus is continuousMathworldPlanetmathPlanetmath. If a<1, then p(1)=1n-a>1-1=0. If a>1, then p(a)=an-a=a(an-1-1)>0. Note also that p(0)=0n-a=-a<0. Thus, if a1, 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,). It follows that u as described in the statement of the theorem exists uniquely. ∎

Title existence of nth 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