We use ultradeep 20 cm data from the Karl G. Jansky Very Large Array and 850 \mu m data from SCUBA-2 and the Submillimeter Array of an 124 arcmin ^ { 2 } region of the Chandra Deep Field-north to analyze the high radio power ( P _ { 20 cm } > 10 ^ { 31 } erg s ^ { -1 } Hz ^ { -1 } ) population . We find that 20 ( 42 \pm 9 % ) of the spectroscopically identified z > 0.8 sources have consistent star formation rates ( SFRs ) inferred from both submillimeter and radio observations , while the remaining sources have lower ( mostly undetected ) submillimeter fluxes , suggesting that active galactic nucleus ( AGN ) activity dominates the radio power in these sources . We develop a classification scheme based on the ratio of submillimeter flux to radio power versus radio power and find that it agrees with AGN and star-forming galaxy classifications from Very Long Baseline Interferometry . Our results provide support for an extremely rapid drop in the number of high SFR galaxies above about a thousand solar masses per year ( Kroupa initial mass function ) and for the locally determined relation between X-ray luminosity and radio power for star-forming galaxies applying at high redshifts and high radio powers . We measure far-infrared ( FIR ) luminosities and find that some AGNs lie on the FIR-radio correlation , while others scatter below . The AGNs that lie on the correlation appear to do so based on their emission from the AGN torus . We measure a median radio size of 1 \farcs 0 \pm 0.3 for the star-forming galaxies . The radio sizes of the star-forming galaxies are generally larger than those of the AGNs .