# Weird pattern in exponentiation

1. Sep 11, 2004

### 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:

$$(x+y)^z \ x^{(y+z)} \ (x \cdot y)^z \ x^{(y \cdot z)} \ (x^y)^z \ x^{(y^z)}$$

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:

$$(x+y)^z \ \ \ \ \ \ \ \ \ x^{(y+z)} \ \ \ \ \ \ \ \ \ (x \cdot y)^z \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^{(y \cdot z)} \ \ \ \ \ \ \ \ \ (x^y)^z \ \ \ \ \ \ \ \ \ x^{(y^z)}$$

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. $$(x+y)^z$$ has the largest identity expression, in terms of the size of the written formula: the binomial theorem. $$x^{(y+z)}$$ and $$(x \cdot y)^z$$ are equal to $$x^y \cdot x^z$$ and $$x^z \cdot y^z$$ respectively, so the shape of their written formulas are identical. $$x^{(y \cdot z)}$$ is equal in value to $$(x^y)^z$$. And finally, $$x^{(y^z)}$$ 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. Sep 11, 2004

### 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.

3. Sep 11, 2004

### 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.

4. Sep 12, 2004

### 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.