We can find two irrational numbers x and y to make xy rational,true or false

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Discussion Overview

The discussion revolves around the question of whether it is possible to find two irrational numbers \(x\) and \(y\) such that \(x^y\) is rational. Participants explore examples and counterexamples related to this concept, engaging in both theoretical and mathematical reasoning.

Discussion Character

  • Exploratory, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant poses the question of whether two irrational numbers \(x\) and \(y\) can yield a rational result when evaluated as \(x^y\).
  • Another participant introduces the term "reasonable numbers" and questions its definition, indicating a potential misunderstanding or ambiguity in terminology.
  • A participant presents an example using \(x = y = \sqrt{2}\) to illustrate that either \(x^y = \sqrt{2}^{\sqrt{2}}\) is rational or irrational, leading to a conclusion that supports the initial claim if either case holds.
  • Another example is provided where \(\sqrt{2}^{\log_2 9} = 3\), further supporting the idea that two irrational numbers can produce a rational result.

Areas of Agreement / Disagreement

There appears to be a general agreement among some participants that it is possible to find two irrational numbers such that \(x^y\) is rational, as demonstrated by the examples provided. However, the discussion includes questions about terminology and the nature of "reasonable numbers," indicating some unresolved aspects.

Contextual Notes

Some participants express confusion regarding the term "reasonable numbers," which may affect the clarity of the discussion. The examples provided rely on specific properties of irrational numbers and logarithms, which may not be universally accepted or understood without further clarification.

Albert1
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we can find two irrational numbers $x$ and $y$
to make $x^y$ rational,true or false statement?
if true then find else prove it .
 
Last edited:
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Albert said:
we can find two unreasonable numbers $x$ and $y$
to make $x^y$ reasonable,true or false statement?
if true then find else prove it .

what is a reasonable number ?
 
kaliprasad said:
what is a reasonable number ?
sorry it should be edited as :
$x,y $ irrational numbers
$ x^y$ rational number
 
Here's a famous example. Let x= y= \sqrt{2}. Either x^y= \sqrt{2}^\sqrt{2} is irrational or it is rational. If it is rational we are done. If it is irrational, let x= \sqrt{2}^\sqrt{2} and y= \sqrt{2}. Then x^y= (\sqrt{2}^\sqrt{2})^\sqrt{2}= \sqrt{2}^{(\sqrt{2}\sqrt{2})}= \sqrt{2}^2= 2. In either case, there exist two irrational numbers, x and y, such that x^y is rational.
 
Another example is $\sqrt{2}^{\log_29}=3$.
 

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