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Math Help - Prove rational raised to a rational is rational.

  1. #1
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    Prove rational raised to a rational is rational.

    "Prove or disprove that if a and b are rational numbers, then a^b is also rational."

    a = s/r, b = c/d.

    (s/r)^(c/d)

    (s^(c/d)) / (r^(c/d))

    Now I am back where I started. Both the numerator and denominator is a rational number raised to a rational number. I'm not even sure If I am taking the right approach. Please help, thank you.
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  2. #2
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    I think you have finished your proof.
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  3. #3
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    Is \displaystyle \left(\frac{1}{2}\right)^{\frac{1}{2}} rational?
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    Quote Originally Posted by jsel21 View Post
    "Prove or disprove that if a and b are rational numbers, then a^b is also rational."

    a = s/r, b = c/d.

    (s/r)^(c/d)

    (s^(c/d)) / (r^(c/d))

    Now I am back where I started. Both the numerator and denominator is a rational number raised to a rational number. I'm not even sure If I am taking the right approach. Please help, thank you.
    I think the final key here is to note that s and r aren't merely rational numbers as you say, but they are integers. And as Prove It so sagely demonstrated, integers to rational powers are not always rational.

    -Dan
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  5. #5
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    If you're not getting anywhere with a proof, it might mean that you haven't hit on the right idea, or it could mean that there can be no proof because the statement is false. See if you can find a counterexample-maybe a certain integer* raised to the 1/2 power.

    *You've at least proved something: if the statement is false, then there must be a counterexample with an integer as the base.

    Looks like Prove It beat me to it, so I'll just add my favorite smiley to serve as my Q.E.D symbol.
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