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Math Help - whitch is larger

  1. #1
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    whitch is larger

    2^{3^{2^{3^{2^3}}}} or 3^{2^{3^{2^{3^2}}}}
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  2. #2
    Senior Member topher0805's Avatar
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    Recall that:

    (x^a)^b = x^{ab}

    So simply multiply all of the exponents out to simplify your expression:

    <br />
2^{3^{2^{3^{2^3}}}} = 2^{108}<br />

    and:

    <br />
3^{2^{3^{2^{3^2}}}} = 3^{72}<br />

    Now simply punch the two into your calculator.
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  3. #3
    Moo
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    Hello,

    I think that the powers are not with the parenthesis.

    According to what he really wrote, this means that 2^{3^2} = 2^9, not (2^3)^2 = 2^6. It should be a bit more complicated
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  4. #4
    Senior Member JaneBennet's Avatar
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    2^{3^{2^3}}=2^{3^8}=2^{6561}

    3^{2^{3^2}}=3^{2^9}=3^{512}

    2^{6561} is a very much bigger number than 3^{512}, and since \ln{3}>\ln{2}, therfore 2^{6561}\ln{3} is a very much bigger number than 3^{512}\ln{2}. On the other hand, \ln{(\ln{2})}<\ln{(\ln{3})} but the difference is minuscule in comparison. Therefore, I think we can conclude that

    2^{6561}\ln{3}+\ln{(\ln{2})}\ >\ 3^{512}\ln{2}+\ln{(\ln{3})}

    that is to say,

    2^{3^{2^{3^{2^3}}}}\ >\ 3^{2^{3^{2^{3^2}}}}
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