"Naše dítě je mnohem chytřejší než vaše!"

Translation: Our child is much smarter than yours!

[vor23]

Why is "mnohem" used here instead of "mnoho"? I thought "je" called for using the nominative case.

[VladaFu]

It does indeed call for nominative and "chytřejší" is nominative. But "mnohem" is an adverb, it does not have cases.

This is a very typical use of an adverb, when you ask "How?" "To what extent?", it modifies the verb "je". So in this case you use "mnohem" or "o mnoho".

You use "mnoho" when you describe a quantity. "Mám mnoho peněz." "I have a lot of money."