Why abbreviation for "Definition" as "Def^n" (math context)

Solution 1:

The n itself does not stand for a sound. The entire abbreviation “Defⁿ” stands for the sound of the word definition.

Could it be you are thinking of “Defⁿ” as a contraction? A contraction is a word in its own right. For example, isn’t is a word, originally coined by contraction of the words is not.

Abbreviations are not contractions. They are still read as the original word. For example, you might abbreviate singular as “sing.” but you would still pronounce it singular.

There really are no general rules for how to abbreviate or how to write an abbreviation, but some words are almost always abbreviated the same way and it is convenient for the reader if you adopt the common abbreviation. For example it is common in academic works to abbreviate pages as “pp.” But that is not a matter of English spelling or grammar, just written style.

Solution 2:

Assuming you mean "defⁿ" rather that the literal "def^n", then it's not an exponent it's just an abbreviation using a superscript.
Wikipedia's entry on superscript:

Many abbreviations use superscripts, especially historically. […] In handwritten shorthand, many abbreviations are still written this way, such as defⁿ […].

If it's the literal "def^n" then I would think it's a humorous use of "^" alluding to its use indicating a superscript.

Solution 3:

In similar situations, I've seen "defn" ('plain'/abbreviation) or "def'n" (with apostrophe / contraction) very frequently. They are usually pronounced as the full word ("definition"), although occasionally 'contracted'.

I would guess that the superscript version is in (possibly humorous) reference to this. I would not assume it is a standard that will necessarily be understood (especially if you'll be teaching 'across a language barrier'). I would be even more careful about "exp^n" for "explain" / advise against it (I've never seen that one, and would think you meant exponentiation). Although, if you have a whole semester, your students will get used to your style (as a previous answer said).

"Def" or "Def." would also be fine, I think.