admirer vs exponent

admirer and exponent both are nouns.

Word	Noun	Adjective	Verb	Adverb
admirer	Yes	No	No	No
exponent	Yes	No	No	No

As nouns, exponent is a hypernym of admirer; that is, exponent is a word with a broader meaning than admirer:

Other hypernyms of admirer include advocate, advocator, proponent.

admirer (noun)	exponent (noun)
someone who admires a young woman	a mathematical notation indicating the number of times a quantity is multiplied by itself
a person who backs a politician or a team etc.	a person who pleads for a cause or propounds an idea
a person who admires; someone who esteems or respects or approves	someone who expounds and interprets or explains