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Math Help - Irrational Numbers Question

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    Irrational Numbers Question

    Is the set of irrational numbers closed under the operation of exponentiation? That is, for all irrational numbers a and b, is a^b irrational? Why?
    Last edited by thamathkid1729; September 25th 2011 at 03:16 PM.
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    Re: Irrational Numbers Question

    i do not know if \sqrt{2}^{\sqrt{2}} is rational, but if it is not, then certainly (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = (\sqrt{2})^2 = 2 is.
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    Re: Irrational Numbers Question

    Quote Originally Posted by thamathkid1729 View Post
    Is the set of irrational numbers closed under the operation of exponentiation? That is, for all irrational numbers a and b, is a^b irrational? Why?
    http://www.math.sc.edu/~filaseta/gra...h785Notes8.pdf
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    Re: Irrational Numbers Question

    Consider e^{\ ln A} = A. if A is a rational number than  \ln A is irrational. So here you have an example of an irrational number raised to an irrational number yielding a rational number result.
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