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## #1 2005-11-01 15:16:37

Makonorth
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### Difficult Exponent Problem

Someone showed me this problem today and I couldn't think of how to solve it:

x^x^x^x^x = 2

Anyone know how to solve it and if so can you explain how you get to your answer?

## #2 2005-11-02 03:29:15

mathsyperson
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### Re: Difficult Exponent Problem

Is that (((x^x)^x)^x)^x or x^(x^(x^(x^x)))?

I'm just asking that to stall for time, because I have no idea how to start trying to solve it. Probably using logs.

Why did the vector cross the road?
It wanted to be normal.

## #3 2005-11-02 04:58:35

kylekatarn
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### Re: Difficult Exponent Problem

x^x^x^x^x=2

x ≈ 1.425385621 ≈ √2

I think the solution of x^x^x^...........=2 converges into √2.
If someone proved that it would be remarkable.

## #4 2005-11-02 05:26:31

Makonorth
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### Re: Difficult Exponent Problem

That's exactly how it was written.

## #5 2005-11-02 07:32:21

MathsIsFun

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### Re: Difficult Exponent Problem

Well then, you should exponentiate in this order I believe:

x^(x^(x^(x^x)))

In Excel (by trial and error) I get: 1.432694...

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #6 2005-11-02 09:13:16

kylekatarn
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### Re: Difficult Exponent Problem

yes, its very interesting.
As you add up more "^x" terms, the solution get closer to SQRT(2).

Again, I would like to see a demonstration of this (or a counter example)
: )

## #7 2005-11-02 09:23:01

mathsyperson
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### Re: Difficult Exponent Problem

Hmm. It's obvious that once it gets to 2 then it will stay there because (√2)² = 2, but that's not anywhere close to a proof.

Why did the vector cross the road?
It wanted to be normal.

## #8 2005-11-02 10:12:54

kylekatarn
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### Re: Difficult Exponent Problem

It's not a proof, but a start.