[dragonwriter]

The US is (by Constitutional design) both a representative democracy and democratic, federal republic.

“Republic” alone is correct, but overly nonspecific; essentially any government that isn't a hereditary monarchy or similar system where government power isn't private and heritable is a republic.

> The definition of republic is 2000 years old.

Well, sure, there is a definition that old (older, actually), but the one you seem to be using (the colloquial American one referring specifically to a system of elected representatives) is much newer, peculiar to American non-technical usage, and equivalent to “democratic republic” and very close to “representative democracy” (differing only in that the latter can coexist with constitutional monarchy.)