Mfcttrf

I'd like to declare the backtick symbol ` to be the math operation power ^, so that by writing

documentclassarticle
usepackageamsmath
% DeclareMathSymbol`mathbinsymbols^
begindocument
$3`2$
enddocument

it is displayed

I was thinking to use DeclareMathSymbol<symbol><type><sym-font><slot> but I'm not sure there is a slot code for ^ since it is an operation rather than a symbol.

I'd like to avoid using newcommand because otherwise I will have to write 3`2 in order to get the power.

All of this is really not recommended, unless you really know what you are doing and/or is willing to suffer the consequences. In the words of Mr. Cat, this is "living dangerously", and you may end up being dead when someone opens the box ;-)

DeclareMathSymbol is used to select the glyph in a font which will be used to write a specific symbol on the paper. The superscript happens in a much lower level in TeX. Usually, the ^ symbol has the category code (catcode) 7, which is a "superscript token". This means that whenever TeX sees a ^, it will make a superscript. At the simplest, you can have that effect by changing the catcode of a ` to 7:

documentclassarticle
catcode``=7
begindocument
$3`2$ and some `quoted' text
enddocument

however, the code above will raise a Missing $ inserted because now that ` is a superscript character, it is only meant to be used in math-mode. You can redefine ` to mean a "normal" (catcode 12) ` character:

documentclassarticle
renewcommand``
catcode``=7
begindocument
$3`2$ and some `quoted' text
enddocument

and then you won't be able to write a grave accent with, say, D'ej`a vu (the output will be "Déj‘a vu"). Plus, any code which relies on ` being a catcode-12 token will fail miserably.

A arguably better alternative is to make the ` an active character only in math-mode and define it equal to a superscript token:

documentclassarticle
catcode``=active
 globallet`=^
mathcode``="8000
begindocument
$3`2$ and some `quoted' text
enddocument

the output is:

But again, this might break something, somewhere, which relies on ` being a catcode 12 token.