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Associative Property for Power Towers?

  1. Jun 14, 2012 #1
    A power tower (x^^n) is a variable raised to the power of itself n amount of times.

    x^^4 = x^x^x^x
    x^^3 = x^x^x
    x^^2 = x^x
    x^^1 = x

    I was wondering if an associative property for power towers exists.

    Does x^(x^x) equal the same thing as (x^x)^x? Is x^(x^^n) equal to x^^(n + 1)?

    If anybody could prove that the order of the exponents doesn't matter if the exponents are the same, that would be great, but an intuitive reasoning would be great also :)

    e-
    WHOA!! I didn't realize I posted this in the physics forum. If anybody would be able to move it to the general math discussion forum that would be great
     
    Last edited: Jun 14, 2012
  2. jcsd
  3. Jun 14, 2012 #2

    jbriggs444

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    You begin by talking about power towers. But then you ask a question about ordinary exponentiation.

    Does (x^y)^z = x^(y^z)?

    well, does (2^2)^8 = 2^(2^8) ?
    does 4^8 = 2^256 ?
    does 65536 = 1.16 x 10^77 ?
     
  4. Jun 14, 2012 #3
    Power towers ARE exponentiation, only all the exponents are the same.

    Does (2^2)^2 = 2^(2^2)? You tell me.

    -e
    You do have a good point though. In order for there to be an associative property for power towers there must be a commutative property of exponentiation.

    3^27 = 7625597484987
    27^3 = 19683

    Evidently there is no commutative property for exponentiation.

    Hmm.. New question then.. How is a power tower defined?

    x^(x^(x^x))
    or
    ((x^x)^x)^x?
     
    Last edited: Jun 14, 2012
  5. Jun 14, 2012 #4

    jbriggs444

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    Fair point.

    So let's try another example

    Is (3^3)^3 equal to 3^(3^3)?
    Is 9^3 equal to 3^27?
    Is 243 equal to 7625597484987?
     
  6. Jun 14, 2012 #5

    jbriggs444

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    What _is_ true is that (x^x)^x is equal to x^(x*x).
    When x=2, it is true that x^x = x*x.
    But when x=3 it is not true that x^x = x*x
     
  7. Jun 14, 2012 #6

    Nugatory

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    Two is a special case, because (x^x)^x = x^(x^2)
    But in general, exponentiation is not associative.

    An interesting follow-up question: For what if any non-negative values of x will the left-associative tower converge? How about the right-associative one?
     
  8. Jun 14, 2012 #7
    This is an excellent point, thanks for sharing.

    I'm sorry but I'm not sure what you mean by the left and right associative towers.

    --
    Thanks for the help guys, I'm heading to work and I'll revisit the thread when I get back.
     
  9. Jun 14, 2012 #8

    jbriggs444

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    If I'm onto what you're about then...

    1^^n = 1 regardless of x and is associative both ways.

    0^^n = 1 for even n and 0 for odd n if you go with right associativity and don't mind getting into a flame war over the definition of 0^0.

    0^^n = 1 for all n if you go with left associativity and don't mind the flame war.

    There is a voice in my head trying to yell that there is a solution to x^x = x that also makes the tower converge.

    But I'm feeling a bit out of my depth now.
     
  10. Jun 14, 2012 #9

    Nugatory

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    The right-associative tower is:
    (x^(x^(x^(x^.......))))

    The left-associative tower is:
    ((...((x^x)^x)^x)^x)....
     
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