I was looking at the different ways the operations +, *, and exponentiation can work on three natural numbers x, y, and z. I found a weird pattern when the second operation performed is exponentiation. These are the expressions:(adsbygoogle = window.adsbygoogle || []).push({});

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

They are arranged in what I think is the most natural way: from "weakest" to "strongest", in the sense + < * < ^, and exponentiation is more powerful when the bigger number is the power, not the base. (This assumes x, y, z are relatively close in size). Here's 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? I'm really not a crackpot, I think there is something here that needs at least some basic explanation, but if someone can explain logically why I'm wrong, I'd be more than willing to accept it.

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# Strange pattern in exponents

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