Prove the irrationality of cbrt[6] - cbrt[5] (I've tried contradiction or finding a poly, but...)

[MathEnjoyer13]

Hello I’ve been doing the following problem:

Prove that ∛6 - ∛5 is irrational

I tried many different approaches like proving by contradiction or trying to get polynomial to apply rational root theorem but nothing works. Can you help me? Thanks in advance

 

[stapel]

Prove that ∛6 - ∛5 is irrational

I tried many different approaches like proving by contradiction or trying to get polynomial to apply rational root theorem but nothing works. Can you help me? Thanks in advance

Please reply *showing* what you've tried, so we can see what's going on. (Read Before Posting)

When you reply, please let us know what topic you're studying (calculus? number theory? proofs?) and recent topics of study, as this can help us guide our assistance.

Thank you!

 

[Steven G]

If a number raised to an integer power is irrational, then is the number itself irrational???