# Exponent of an exponent?

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.

• en.wikipedia.org/wiki/Operator_associativity says : an exponentiation operator (if present) is right-associative Sep 1, 2015 at 16:16
• Sep 1, 2015 at 16:16
• Generally, we don't write $2\hat{\phantom:}3\hat{\phantom:}4$. We write $2^{3^4}$. And that means $2^{(3^4)}$ unambiguously. Sep 1, 2015 at 18:54

Conventionally $a^{b^c}$ means $a^{(b^c)}$.

The other way of parsing it, $(a^b)^c$, yields a result equal to $a^{bc}$. In particular $(2^3)^4 = 2^{3\times 4} = 2^{12} = 4096$.

• Thank you, I actually had written it the way I did, though, because that was how it had been presented to me in my programming textbook. I was pretty sure it was the conventional format, but couldn't find precedent. Sep 1, 2015 at 18:02
• If it's in a programming language, you need to find out whether the ^ operator in that language is left-associative (x^y^z = (x^y)^z) or right-associative (x^y^z = x^(y^z)). Math doesn't use ^ as the exponentiation operator.
– Joe
Sep 2, 2015 at 6:57
• Funny that this answer is the accepted one when it actually doesn't answer the question, which is about a^b^c.
– user65203
Sep 2, 2015 at 9:39

There are a number of type-setting situations with algebraic expressions that cause problems when you try to enter them into a one-line input calculator.

For example $$\frac{2+3}{4+5}$$must be entered into a calculator as $$(2+3)/(4+5)$$The horizontal line in the fraction implies brackets around the expression in the numerator and denominator.

Similarly, the expression$$\sqrt{2+5}$$must be entered on my Casio as$$\sqrt{}(2+5)$$Here the horizontal line above the expression shows what expression is to be included in the root; the brackets in the second version do the same thing for the calculator.

Finally, if an exponent is itself an expression, that typesetting defines exactly what is in the exponent; if you can't enter the typesetting directly, you need to supply the missing information.$$2^{3^4}$$should be entered into a calculator as $$\text{2 ^ ( 3 ^ 4 )}$$

Note that when you enter 2 ^ 3 ^ 4 into Wolfram Alpha, it back translates the expression into the typeset version and then evaluates that expression correctly

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.

• Java, it's for my Data Structures class. My professor is flabbergasted as well. Sep 2, 2015 at 1:31
• @FalconicKatadora Java?! But Java uses ^ for another purpose, so your question is not really relevant there. See updated answer above. Sep 2, 2015 at 9:27
• It's in the textbook, not in a code snippet though. Sep 2, 2015 at 11:57
• @FalconicKatadora What is the context in the book? Why would they use ^ notation for exponentiation in a textbook on Java? Sep 2, 2015 at 12:14
• It is in reference to infix/polish/RPN. Other than that I have no clue as to why this author would do such... This author has been known to make really odd choices in his code too though. Sep 3, 2015 at 12:56