Origins
The English word "exponent" derives from Latin "expōnēns," the present participle of "expōnere," meaning to put forth, set out, or explain. The Latin verb combines "ex-" (out) with "pōnere" (to put, to place).
The word entered English in the 16th century with its general meaning: a person who sets forth or advocates ideas. The mathematical sense — a superscript number indicating repeated multiplication — followed in the 17th century, building on the metaphor of a number that 'sets forth' the power of another.
Latin Roots
Latin "pōnere" is one of the most productive roots in English. Through various Latin prefixes, it generated an enormous family: "position" (a placing), "compose" (to put together), "propose" (to put forward), "dispose" (to put apart), "oppose" (to put against), "expose" (to put out), "deposit" (to put down), and "component" (putting together). Each preserves the core concept of placing or putting, modified by its prefix.