Associative Property for Power Towers?

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Discussion Overview

The discussion revolves around the associative property of power towers, specifically whether expressions like x^(x^x) and (x^x)^x yield the same results. Participants explore the implications of exponentiation in the context of power towers, questioning definitions and properties related to their structure and behavior.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether the associative property holds for power towers, specifically if x^(x^x) equals (x^x)^x.
  • Others point out that power towers are a specific form of exponentiation where all exponents are the same, leading to discussions about the general properties of exponentiation.
  • A participant raises the issue of commutativity in exponentiation, citing examples that demonstrate it does not hold.
  • There is a discussion about how power towers can be defined, with different interpretations of their structure being proposed.
  • Some participants note that for certain values of x, such as 2, specific equalities hold, while for others, like 3, they do not.
  • A later reply introduces the concept of left and right associative towers, prompting further clarification on their definitions and convergence behavior.
  • One participant mentions special cases for the values of 1 and 0 in the context of power towers and their associativity.

Areas of Agreement / Disagreement

Participants express differing views on the associative property of power towers, with no consensus reached. Some agree on specific cases where properties hold, while others challenge these claims, leading to an unresolved discussion.

Contextual Notes

Limitations include varying definitions of power towers and the implications of associativity and commutativity in exponentiation. The discussion also highlights the complexity of convergence in left versus right associative towers.

Hertz
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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:
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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 ?
 
jbriggs444 said:
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 ?

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:
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?
 
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
 
Hertz said:
Power towers ARE exponentiation, only all the exponents are the same.

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

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?
 
jbriggs444 said:
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

This is an excellent point, thanks for sharing.

Nugatory said:
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?

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

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

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

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