weird pattern in exponentiation

StatusX

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?

HallsofIvy

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.

StatusX

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.

StatusX

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.