My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum

Thanks Tree4Thanks
  • 4 Post By romsek
LinkBack Thread Tools Display Modes
February 2nd, 2018, 07:32 PM   #1
Senior Member
Joined: Oct 2015
From: Antarctica

Posts: 128
Thanks: 0

Exclamation Proof of uniqueness of positive cubes

Prove that there exists exactly one positive $t\in \mathbb{R}$ with $t^3=c$ for $c>0$.

I assume that most of my derivation needs to be algebraic and using some simple axioms. So for example, if I define the set:

$$T:=\lbrace t\in \mathbb{R}: t> 0, t^3 \leq c \rbrace$$

I must show that the cubed root of $c$ is the supremum of $T$ and then prove its uniqueness. But I'm not sure how to go about this. How can I concisely prove this?
John Travolski is offline  
February 2nd, 2018, 08:42 PM   #2
Senior Member
romsek's Avatar
Joined: Sep 2015
From: USA

Posts: 2,038
Thanks: 1063

suppose there are $t_1\neq t_2$ such that $t_1^3=t_2^3=c$


$t_1^3 - t_2^3 = 0$

$(t_1 - t_2)(t_1^2 + t_1 t_2 + t_2^2) = 0$

well we've assumed that $t_1 \neq t_2 $ so it must be that

$t_1^2 + t_1 t_2 + t_2 ^2 = 0$

$\left(t_1 - \dfrac{t_2}{2}\right)^2 + \dfrac{3 t_2^2}{4} = 0$

It should be pretty clear that this has no real solutions.

Thus the original assumption is incorrect and $t_1 = t_2$
romsek is offline  

  My Math Forum > College Math Forum > Real Analysis

cubes, positive, proof, uniqueness

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Proof that i is poth positive and negative Tau Math 40 December 31st, 2016 06:23 PM
Proof uniqueness of the nth root annakar Number Theory 1 December 4th, 2012 11:06 AM
Graph theory proof (uniqueness of minimum spanning tree) aeromantang Applied Math 3 December 28th, 2011 10:34 AM
Uniqueness of Square Root proof jstarks4444 Number Theory 2 November 16th, 2010 12:19 PM
X is a random variable with positive value,proof that E[X]=. johnmath Advanced Statistics 1 July 24th, 2010 06:56 PM

Copyright © 2018 My Math Forum. All rights reserved.