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Irrational^rational = rational

  1. Feb 16, 2014 #1
    Is it true that in some cases an irrational number raised to a rational power is rational? If so what kinds of irrational numbers?
     
  2. jcsd
  3. Feb 16, 2014 #2

    jgens

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    Yes. For your first question note that √22 = 2. For your second question all number of this form should be algebraic.

    Edit: As Mark44 pointed out below I need to add the caveat that the exponent be non-zero for my second claim to hold.
     
    Last edited: Feb 16, 2014
  4. Feb 16, 2014 #3

    pwsnafu

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    It's also worth pointing out that irrational to the irrational can be rational.

    The Gelfond-Schneider theorem says: if ##a## and ##b## are algebraic, with ##a \neq 1,0## and ##b## irrational, then ##a^b## is transcendental.

    This means that ##\sqrt{2}^{\sqrt{2}}## is transcendental (hence irrational). Raise this to the power of ##\sqrt{2}## and you get 2.
     
  5. Feb 16, 2014 #4

    Mark44

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    A very simple example is ##\pi^0 = 1##.
    ##\pi## is irrational, and 0 is rational.
     
  6. Mar 13, 2014 #5
    if the order of the root is say x, and the power you raise are multiples of x , then it becomes a rational number(or if the power is 0)
    example:√2^4=4
     
  7. Mar 13, 2014 #6

    HallsofIvy

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    Note that the irrational number has to be algebraic. A transcendental number raised to a rational power cannot be rational. In fact, it must still be transcendental.
     
  8. Mar 13, 2014 #7

    DrClaude

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    Isn't there a contradiction here?
     
  9. Mar 13, 2014 #8

    arildno

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    I think you just became a candidate for the Fields medal! :smile:
     
  10. Mar 13, 2014 #9

    DrClaude

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    No , I just became the candidate for another coffee!

    Sorry Mark44, completely missed your humor there...
     
  11. Mar 13, 2014 #10

    Mark44

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    That was arildno...
     
  12. Mar 13, 2014 #11

    DrClaude

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    I thought you were trying to be funny by proposing the trivial case "to the power of 0". :redface:
     
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