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Weird pattern in exponentiation

  1. Sep 11, 2004 #1

    StatusX

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    I was looking at the different ways the operations +, *, and exponentiation can work on three numbers x, y, and z. I found a weird pattern when the second operation performed is exponentiation. These are the expressions:

    [tex] (x+y)^z \ x^{(y+z)} \ (x \cdot y)^z \ x^{(y \cdot z)} \ (x^y)^z \ x^{(y^z)} [/tex]

    Notice how I arranged them in a natural way, where the first operation(inside the parantheses) is (+,+,*,*,^,^), and the second operation, exponentiation, is carried out on the (R,L,R,L,R,L) of the parantheses. Now look at the pattern:

    [tex] (x+y)^z \ \ \ \ \ \ \ \ \ x^{(y+z)} \ \ \ \ \ \ \ \ \ (x \cdot y)^z \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^{(y \cdot z)} \ \ \ \ \ \ \ \ \ (x^y)^z \ \ \ \ \ \ \ \ \ x^{(y^z)} [/tex]

    largest ......<- identical ->....... | .......<- indentical ->...... largest

    written ........ written ............ | ............ value .............. value
    formula ........ formula

    I'm sorry if this doesn't format right, but I'll explain what it means. [tex] (x+y)^z[/tex] has the largest identity expression, in terms of the size of the written formula: the binomial theorem. [tex] x^{(y+z)}[/tex] and [tex] (x \cdot y)^z[/tex] are equal to [tex] x^y \cdot x^z [/tex] and [tex] x^z \cdot y^z [/tex] respectively, so the shape of their written formulas are identical. [tex] x^{(y \cdot z)} [/tex] is equal in value to [tex] (x^y)^z[/tex]. And finally, [tex] x^{(y^z)}[/tex] has the largest value, for x,y,z>>1.

    This seems like a very bizzare link between the "man-made" (sort of) written formulas and the "completely natural" values of these expressions. Is there anything to this, or is it just a coincidence?
     
  2. jcsd
  3. Sep 11, 2004 #2

    HallsofIvy

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    I have no idea why you consider your arrangement to be "in a natural way". There would be absolutely no difference that I can see if you were to arrange them in any other way.
     
  4. Sep 11, 2004 #3

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    They are arranged regularly. You might argue if its natural or not, although I'm pretty sure they are in order of increasing value for x,y,z >>1, which seems pretty natural.
     
  5. Sep 12, 2004 #4

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    So does this need to be explained, or am I reading too much into it? I could see how you might argue the arrangement is arbitrary, but its at least in increasing order of the "power" of the first operation, ie., (+,+),(*,*),(^,^). Then the only choice I made that may seem arbitrary is which side the exponent should be on in the first of each pair, and I picked the right side. But like I said, I also think they are in order of value for numbers >>1 (maybe just >2?), but I'm not completely sure about that.
     
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