What is x^x^x?

In my experience, chained exponents are indeed evaluated from right to left (the opposite of the usual order), precisely for the reason you alluded to -- for any number x, x^x^x^x^x... (n times) can be written as x^(x^(n-1)) if we assume left-to-right evaluation, and only the right-to-left evaluation requires an actual exponent chain. Thus, from the standpoint of minimizing the number of required parentheses, it makes the most sense to evaluate from right-to-left, because otherwise x^x^x^x^x... (n times) could not be written without the use of ungainly numbers of parentheses (i.e. x^(x^(x^(x^x)))), for n=5). Further, this usage also reflects better the evaluation of chained exponents in print, where normally the subsequent x's would be written as superscripts of the superscript, and thus unambiguously intended to be exponents of the exponent, rather than exponents of the entire preceding expression.

That said, you won't be able to find a "proof" of the rule for chained exponents, because like all rules regarding order of operation, it is merely a social convention we use to simplify some expressions, not an expression of some underlying mathematical fact. That said, if you're looking for an ex cathedra pronouncement of this rule, you might quote the wikiepdia article on tetration:

"Note that when evaluating multiple-level exponentiation, the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words:

2^2^2^2 = 2^(2^(2^2))) = 2^(2^4) = 2^16 = 65536

2^2^2^2 is _not_ equal to ((2^2)^2)^2 = (4^2)^2 = 16^2 = 256

(This is the general rule for the order of operations involving repeated exponentiation.)"

It is important that you follow convention, because you get differnent answers to x^x^x depending on which of the ^-signs you use up first:

Consider 3^3^3..

If we do 3^(3^3) we get 3^27 {because brackets should be done first}, and

3^27 = 7,625,597,484,987

but, if we do (3^3)^3, we get 27^3 = 19,683

My TI-83 calculator gives me 19,683 when I enter 3^3^3, so I would go with that answer.

The advantage of doing it this way, is that if you had to calculate a^b^c^d^. . . . ., you could at least start working it out by doing a^b, then (a^b)^c, then {(a^b)^c}^d. . . and so on, whereas, if the powers went to infiity, you couldn't make a start to the evaluation, . . .even if it actually gave a finite answer !

I will be honest with you - I don't know whether there is a "definitive" answer. There should be, because the rules in maths have been introduced to avoid ambiguity.

Of course, there is a way round it. You could define x^x^x to mean (x^x)^x, or, you could answer your qustion both ways.

You could say: if y = x^x^x = (x^x)^x then . . .dy/dx = etc.

but, if y = x^x^x = x^(x^x) then dy/dx = ....

You might be criticised for hedging your bets, but at least you will have demonstrated that you were able to differentiate y no matter how it was meant to be interpreted !

In the absense of clarifying parentheses,

x^x^x

should be interpreted from right to left. Here is a link to a page on MathWorld about Fermat Primes that uses that interpretation.

http://mathworld.wolfram.com/FermatPrime.html

Puggy, you're simply NOT consistent!

1. You chided someone earlier for writing "solve" when they meant "evaluate," and now you're doing the VERY SAME THING.

2a. You are also EXTREMELY confusing when you write that x^x^x could be interpreted as x^x^2 if "worked left to right," which you defined and therefore in this context presumably interpreted as (x^x)^x. The expression x^x^2 (your supposed "left to right" final interpretation) would in that case [as (x^x)^2] involve first four x's and then with further simplification two x's: (x^x)*(x^x) = x^ (x + x) = x^(2x), and so on.

2b. In fact you seem to have switched your own parenthetical horses in mid-stream: worked "left to right," you're saying that x^x^x [that would be, (x^x)^x] would be x^x^2, but it would ONLY BE THAT if you interpreted the LATER EXPRESSION inconsistently, that is in the ABSOLUTELY OPPOSITE way, namely as x^(x^2) ! See?! --- explanatory horses switched in mid-stream!

Consider the following obvious point. WITHOUT the PROPER USE of parentheses or brackets, CONFUSION is BOUND to reign! Just look at:

(3^3)^3 = (27)^3 = 19683; but

3^(3^3) = 7.62559748.. x 10^12

It is NOT worth having a convention so that mathematical illiterates can avoid the simple and straightforward task of using parentheses or brackets to write unambiguous mathematics instead of mathematical gibberish! Without the use of them, chaos can rule. With them, all is clear.

The convention should be: use parentheses or brackets WHEREVER possible confusion would otherwise arise. The onus is on the WRITER to write things properly, NOT upon the READER to interpret confused thinking and expression. No further convention should be introduced, simply to avoid writing things properly.

It's a little like using physics to explain things out there in the larger universe: no NEW physics should be introduced until you've exhausted what can be understood by already known physics.

Live long and prosper.

POSTSCRIPTS :

A.) It has been suggested that the meaning should be that which makes calculator use easy. This matter SHOULD NOT be handed over to the makers or users of calculators. They have already confused mathematical terminology by using an ambiguous ' TAN^(-1) ' on their buttons --- at least on my 'Sharp' calculator. This is ambiguous because there was long a mathematical convention that ' arctan ' or ' tan^(-1) ' were MULTIVALUED functions of their arguments, while ' Arctan ' (or in some quarters ' Tan^(-1) ') represented the so-called PRINCIPLE VALUES, i.e. only those lying between - π/2 and + π/2. In other words, these multi- or single-valued functions were distinguished by their initial letters (only) being lower or upper case respectively. Perhaps the ' TAN^(-1) ' button on my calculator is in effect adhering to but extending that convention by capitalising the whole function name, but one can't be at all sure because EVERYTHING like SIN, COS, LOG, LN, etc. is capitalised.

(I am grateful to Pascal for pointing out that such buttons don't use ' arctan ' ; but then again, mine doesn't use the lower case ' tan^(-1) that he suggested was common.)

B.) Meanwhile, as another example of calculator confusion, two TI users below ('sumzrfun' and 'rawfulco...') have told us that their own TI calculators work in completely OPPOSITE ways. Could the two of them possibly resolve this manifest discrepancy?!

According to the TI's (ya those silly calculators) syntax which is the syntax that I use for typing equations with just the keyboard, and how i read them

x^x^x is indeed x^(x^x) <------

although whenever I have to right something like that I always put parenthesis just to make sure people don't' get confused

People.

The order of operations is dictated by convention and so tends towards what is most useful.

a ^ b ^ c meaning (a ^ b) ^ c which equals a ^ (b c) is rather useless.

a ^ b ^ c meaning a ^ (b ^ c) is much more useful and is therefore the convention.

For example, that is how Perl assocates its ** operator.

Dan

First x^x^x = x^x^2 is not true. x^(x*x)=x^x^2

I always interpret multiple ^ to indicate starting at the right of the set.

the right way to solve would be ((x)^x)^x.

When differentiating it is easiest to consider this a s a composite function F(F(x)) where F(n) = n^x.

By the way, x=.5 is an example of where it matters what order you select.

It should be interpreted like this

((x)^x))^x