Picture someone taking a thing out of a bag and setting it down on the table where everyone can see it. That act — putting something out in the open so it can be looked at — can be taken as the thread behind exponent. A person can set out an idea by explaining it, or set out a skill by performing it, and in maths a small number is set out above the line to show how many times to multiply.
From the word’s history
The source records exponent from Latin expōnēns, the present participle of expōnō (to expose, exhibit, display, set out, explain), itself ex- (out, away) plus pōnō (to lay, place, put). The literal sense of putting something out is what the modern core picture rests on.
資料記的是：exponent 來自拉丁文 expōnēns，是動詞 expōnō 的現在分詞，而 expōnō 由 ex-（向外）加 pōnō（放置）組成，意思包含「展示、擺出來、說明」。今天這個「攤開來給人看」的核心畫面，正是從這個字面來的。
Wiktionary (kaikki:3f6f69a190036ae4:767842254:36136), etymology of exponent (noun)
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