We present the first results from the deep and wide 5 GHz radio observations of the Great Observatories Origins Deep Survey (GOODS)-North ($sigma=3.5 ; mu Jy ; beam^{-1}$, synthesized beam size $theta =$ 1.47 arcsec $times$ 1.42 arcsec, and 52 sources over 109 arcmin$^{2}$) and GOODS-South ($sigma=3.0 ; mu Jy ; beam^{-1}$, $theta=$0.98 arcsec $times$ 0.45 arcsec, and 88 sources over 190 arcmin$^{2}$) fields using the Karl G. Jansky Very Large Array. We derive radio spectral indices {alpha} between 1.4 and 5 GHz using the beam-matched images and show that the overall spectral index distribution is broad even when the measured noise and flux bias are considered. We also find a clustering of faint radio sources around $alpha=0.8$, but only within $S_{5GHz} < 150 ; mu Jy$. We demonstrate that the correct radio spectral index is important for deriving accurate rest frame radio power and analyzing the radio-FIR correlation, and adopting a single value of $alpha=0.8$ leads to a significant scatter and a strong bias in the analysis of the radio-FIR correlation, resulting from the broad and asymmetric spectral index distribution. When characterized by specific star formation rates, the starburst population (58%) dominates the 5 GHz radio source population, and the quiescent galaxy population (30%) follows a distinct trend in spectral index distribution and the radio-FIR correlation. Lastly, we offer suggestions on sensitivity and angular resolution for future ultra-deep surveys designed to trace the cosmic history of star formation and AGN activity using radio continuum as a probe.
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