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Nested Exponents

  1. Nov 22, 2012 #1
    Hello everyone, I have stumbled across a curious question while programming.

    To start the process, say I have a a positive integer greater than one: α0
    Nesting that in an exponent is a simple operation and a common occurrence: α0α1
    1 is also a positive integer greater than 1. In fact, all numbers mentioned in this problem of mine are in that given set)

    But then I run into a problem when I nest this number with yet another exponent: α0α1α2
    See, the exponent for me is calculated from right to left, so I can't simply store α0α1 into memory and then set that to the α2 power, because that only works on operators that are calculated left to right.

    So my question is this: Given a chain of nested exponents α0α1α2...αn, is there a closed form means to determine α0α1α2...αnαn+1 without having to recalculate the exponents from right to left?

    I apologize if this is the wrong place to ask, but as a programmer, nested exponents are causing me a headache, because I have to spend n operations to calculate the nested exponent every time I nest it. While not completely slow, I would enjoy a faster means of computing this. I am also aware my question might not seem completely clear, so please ask for clarifications if you require any.

    Thanks for your time.
  2. jcsd
  3. Nov 22, 2012 #2


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    Try reviewing rules of exponents using some simple example.

    Find [itex]2^{{3}^{2}^{5}}[/itex]. Try doing the exponents from right to left; and also try doing from left to right. See what results.

    format tags not working right. 2^3^2^5
    Using TexAide: [itex]\[
    2^{3^{2^5 } }

    [STRIKE]Should be the same as [itex]2^{3*2*5}[/itex][/STRIKE]Obviuos mistake. That would not be result of the nesting. See next member's post.
    Last edited: Nov 22, 2012
  4. Nov 22, 2012 #3


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    By nesting the OP means 2^(3^4) = 2^81 = 2417851639229258349412352, this is different than (2^3)^4 = 8^4 = 4096.
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