# Is there a more correct way to show explicit exponential operators?

I typed in this question like this:

$\underbrace{\alpha \hat{\phantom{\hat{}}} \alpha \hat{\phantom{\hat{}}} \dots \hat{\phantom{\hat{}}} \alpha}_\text{n times}$


which mathjax prints as

I feel there should be a more correct way to do this.

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I am not sure that that is such a good idea since exponentiation is not associative: 2^(3^2) is not the same as (2^3)^2). So what exactly does 2^3^2 mean? This is probably the reason why TeX complains about double super scripts. –  Peter Grill Jul 28 '12 at 23:32
@PeterGrill if you look at the linked question, you'll see that the expression is referring to the number of possible results with parentheses inserted in all possible ways. –  Gonzalo Medina Jul 28 '12 at 23:39

You could also type

\def\mhat{\mathbin{\hat{\vphantom{x}}}} % spacing like a binary operator
$\underbrace{\alpha\mhat\alpha\mhat\cdots\mhat\alpha}_{\text{$n$times}}$


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There doesn't seem to be a circumflex (hat) symbol intended for use in maths mode, except as an accent. However, you can use the text mode version \textasciicircum (or alternatively just \^{}). This avoids the need for phantoms.

\documentclass{article}
\usepackage{amsmath}
\newcommand\mathcirc{\text{\textasciicircum}}
\begin{document}
$\underbrace{\alpha \mathcirc \alpha \mathcirc \ldots \mathcirc \alpha}_{n\text{ times}}$
\end{document}

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Perhaps you might like to use

\newcommand\expon{\mathbin {^\wedge}}


or, if you're quite flexible about the symbol you use, adopt Knuth's up-arrow notation:

\newcommand\expon {\mathbin {\uparrow}}

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A variant without hat that counts the exponential operations:

$\bigl(\bigl(\bigl(\alpha \underbrace{ ^\alpha\bigr)^\alpha\bigr)^{\cdots}\bigr)^\alpha }_\text{$n$times} = \alpha^{(\alpha^n)}$


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But the idea of the OP is not to include the parentheses explicitly; the OP is defining an expression for counting all possible different values obtained by inserting parentheses in every possible way in the original expression. –  Gonzalo Medina Jul 29 '12 at 0:38
I think that it would be better to add centered dots after the first parenthesis : \bigl(\cdots\bigl(\bigl(.... –  projetmbc Jul 29 '12 at 10:44

Just like Heiko's answer, but without the parentheses:

\alpha
\underbrace{
{{{^{\alpha\vphantom{h}}}^{\alpha\vphantom{h}}}^{\cdots\vphantom{h}}}^{\alpha\vphantom{h}}
}_{\text{$n$ times}}
= \alpha^{(\alpha^n)}


(A \strut instead of \vphantom{h} gives me too much vertical space.)

(Probably better with \! after the initial \alpha.)

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This unfortunately misses the point of the OP in the same way as Heiko's answer; with the additional problem that the equation you write is incorrect. –  Niel de Beaudrap Jul 29 '12 at 9:47
The notation without parenthesis is very ambiguous. –  projetmbc Jul 29 '12 at 10:45
@projetmbc But wasn't it the point? I thought the OP wanted an expression where parentheses could be inserted in every possible way. (Of course, I copied the equation from Heiko, so the equality only holds for a particular way of inserting the parentheses, but I thought the LHS is what the OP wanted to express.) –  Jellby Jul 29 '12 at 10:59
@NieldeBeaudrap: As Jellby just explained, only the LHS is important here. (Ok, then maybe he should remove the RHS). Any way, I think this is a great way of writing the expression, as you don't need to define a new exponentiation operator. –  canaaerus Jul 29 '12 at 13:33
Umm... When you have an actual vertical tower of exponents, the fact that there are no parentheses doesn't make it ambiguous. The exponents are then evaluated top-down. For instance. If \alpha=2 and n=3, the result would be 16 (or 2^4). That's standard mathematical notation. It's when you introduce an explicit, non-associative boolean symbol as in the OP that ambiguity arises. –  Niel de Beaudrap Jul 30 '12 at 10:57