Active galaxies are frequently considered to be powered primarily either by an ionizing compact (quasar) core or else by starburst activity. Long-baseline radio observations provide an acute tool for discriminating between these two mechanisms, and we have applied this technique to samples of Seyfert, starburst, and extremely luminous far-infrared galaxies (ELFs). We find that (1) radio cores occur in about one-third of optically selected Seyferts, but are rare in optically selected starburst galaxies, confirming that the Seyferts are intrinsically different from starbursts, and (2) radio cores occur in about one-third of ELFs with Seyfert-like spectra but are rare otherwise. This implies that some ELFs are powered by quasar cores (and generally betray themselves as such by a Seyfert-like spectrum) but that others, including some of the most luminous ones, are not, and therefore require some other mechanism, such as superstarburst activity, to explain them.