Associative Property for Power Towers?

In summary: And yes, you are correct that for 0^0, there is no clear solution and it can lead to a flame war. The left-associative tower definition may be more widely accepted, but it ultimately depends on the context and how the tower is being used.
  • #1
Hertz
180
8
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
 
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  • #2
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 ?
 
  • #3
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?
 
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  • #4
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?
 
  • #5
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
 
  • #6
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?
 
  • #7
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.
 
  • #8
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.
 
  • #9
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)...
 

1. What is the Associative Property for Power Towers?

The Associative Property for Power Towers is a mathematical property that states that when multiplying a power tower (a number raised to a power, raised to a power, and so on) by another number, the order in which the powers are multiplied does not affect the outcome. In other words, (a^b)^c is equal to a^(b^c).

2. Why is the Associative Property for Power Towers important?

The Associative Property for Power Towers is important because it allows us to simplify complex power tower expressions. By rearranging the order of the powers, we can make the calculations easier and more manageable.

3. Can the Associative Property for Power Towers be applied to any power tower expression?

Yes, the Associative Property for Power Towers can be applied to any power tower expression, as long as the numbers involved are real numbers and the operations are multiplication. This property does not apply to addition or subtraction.

4. How is the Associative Property for Power Towers different from the Commutative Property?

The Associative Property for Power Towers and the Commutative Property are both mathematical properties that involve changing the order of operations. However, the Commutative Property states that the order of operations does not affect the outcome, while the Associative Property for Power Towers specifically applies to power tower expressions.

5. Are there any exceptions to the Associative Property for Power Towers?

No, there are no exceptions to the Associative Property for Power Towers. This property holds true for all real numbers and power tower expressions. It is a fundamental property of multiplication that cannot be broken.

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