TITLE: Exponent of an exponent?
QUESTION [8 upvotes]: If I have an expression that gives 2^3^4, would I compute this as $(2^3)^4$ or as $2^{(3^4)}$? The two answers are wildly different.
My TI gives the former but Wolfram gives the latter and I don't know which to trust more on math. I also tried reading up in textbooks on the Laws of Exponents, but I could not find the case A^B^C in any of the textbooks I can access...
EDIT: This was found as an example in a CompSci textbook, thus the notation as it was.

REPLY [3 votes]: In standard mathematics the notation $A \mathrel{^\wedge} B$ is not used; one does $A^B$ instead.
In some computer science (programming) applications the infix circumflex operator notation $A \mathrel{^\wedge} B$ is used (or rather the ASCII version A^B). What you point out is that this operator $\mathrel{^\wedge}$ is not associative.
So does $A \mathrel{^\wedge} B \mathrel{^\wedge} C$ mean $A \mathrel{^\wedge} (B \mathrel{^\wedge} C)$ (that is $A^{B^C}$) or $(A \mathrel{^\wedge} B) \mathrel{^\wedge} C$ (which is $(A^B)^C$)? Well, that depends on the language! For example in PARI/GP and Wolfram Alpha it would be the former, while in Visual Basic .NET and some calculators it would be the latter.
So if you are writing a computer program, find out what convention the programming language in question uses. If you are writing for a human reader, avoid $A \mathrel{^\wedge} B \mathrel{^\wedge} C$ without parenthesis because it can be ambiguous.
For similar reasons, avoid $A/B/C$ and use either $A/(B/C)$ or $(A/B)/C$. Of course, whenever it is technically possible, use "real" mathematical notation like $\frac{A}{\frac{B}{C}}$ or $\frac{\frac{A}{B}}{C}$.
PS! Out of curiosity, was the textbook using a particular programming language?

Addition: In the Java programming language, A^B is not exponentiation but instead bitwise exclusive OR (bitwise XOR). See What does the ^ operator do in Java? Since this operation is associative, A^(B^C) == (A^B)^C for XOR, there will usually not be an issue here.