Irrational Number Raised To Irrational Number

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An irrational number raised to an irrational power can indeed yield a rational result, as demonstrated by the example A = (sqrt{2})^(sqrt{2}). If A is rational, the problem is resolved; if A is irrational, raising it to the power of sqrt{2} results in a rational number, specifically 2. The discussion emphasizes the elegance of this mathematical concept and the non-constructive proof approach. Participants express uncertainty about proving A's irrationality but agree on the validity of the conclusion. The thread highlights the beauty and complexity of irrational numbers in mathematics.
  • #31
PeroK said:
Or, perhaps, ##3^{p/q} = 2##?
Ah, yes, let me edit. Edit: Edited.
 

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