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Math Help - Irrationality

  1. #1
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    Irrationality

    Question:
    For irrational numbers x,y show that x^y is not necessarily irrational.

    How would I go about this question? I thought about proof by contradiction but don't think it will work because x^y could also be irrational. Do I just find a counter-example?
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  2. #2
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    Quote Originally Posted by Gusbob View Post
    Do I just find a counter-example?
    This seems to be a convenient way to go about answering the question.

    Solving x^{\sqrt{2}} = 2 for x:

    x = 2^{\frac{1}{\sqrt{2}}} = 2^{\frac{\sqrt{2}}{2}} = \sqrt{2^{\sqrt{2}}}

    Now you just have to show that \sqrt{2^{\sqrt{2}}} is irrational.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Gusbob View Post
    Question:
    For irrational numbers x,y show that x^y is not necessarily irrational.

    How would I go about this question? I thought about proof by contradiction but don't think it will work because x^y could also be irrational. Do I just find a counter-example?
    I'm not entirely sure what you're entitle to assume. Easies example 2=e^{\ln(2)}
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  4. #4
    Super Member Bacterius's Avatar
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    Statement : For irrational numbers x,y show that x^y is not necessarily irrational.

    This statement is equivalent to disproving the statement : For irrational numbers x,y show that x^y is necessarily irrational.

    Therefore, one counter-example (such as the one given by Drexel) is enough to disprove the latter, and thus to prove the original statement.
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