(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove [itex]5^{1/3} - 3^{1/4}[/itex] is irrational.

2. Relevant equations

http://www.purplemath.com/modules/solvpoly.htm

3. The attempt at a solution

Ok, what I have tried doing is using the about Rational Roots property by letting [itex]x = 5^{1/3} - 3^{1/4}[/itex] and trying to pull out a polynomial where we can see the possible Rational Roots. If the number above is not in the list, then it is irrational.

My problem is that, is this really the way to go? It feels like I'm going to get a polynomial of degree 30 or something. That's fine, but getting there is a *****.

Also, the assume [itex]5^{1/3} - 3^{1/4}[/itex] is rational and let [itex]p/q = 5^{1/3} - 3^{1/4}[/itex] seems hopeless too.

Anyways, what approach is the most practical?

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# Proving something is irrational.

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