We present nearly simultaneous Chandra and NuSTAR observations of two actively star-forming galaxies within 50 Mpc: NGC 3256 and NGC 3310. Both galaxies are detected by both Chandra and NuSTAR, which together provide the first-ever spectra of these two galaxies spanning 0.3-30 keV. The X-ray emission from both galaxies is spatially resolved by Chandra; we find that hot gas dominates the E < 1-3 keV emission while ultraluminous X-ray sources (ULXs) dominate at E > 1-3 keV. The NuSTAR galaxy-wide spectra of both galaxies follow steep power-law distributions with Gamma ~ 2.6 at E > 5-7 keV, similar to the spectra of bright individual ULXs and other galaxies that have been studied by NuSTAR. We find that both NGC 3256 and NGC 3310 have X-ray detected sources coincident with nuclear regions; however, the steep NuSTAR spectra of both galaxies restricts these sources to be either low luminosity AGN or non-AGN in nature (e.g., ULXs or crowded X-ray sources that reach L2-10 keV ~ 10^40 erg/s cannot be ruled out). Combining our constraints on the 0.3-30 keV spectra of NGC 3256 and NGC 3310 with equivalent measurements for nearby star-forming galaxies M83 and NGC 253, we analyze the SFR-normalized spectra of these starburst galaxies. The spectra of all four galaxies show sharply declining power-law slopes above 3-6 keV due to ULX populations. Our observations therefore constrain the average spectra of luminous accreting binaries (i.e., ULXs). This result is similar to the super-Eddington accreting ULXs that have been studied individually in a targeted NuSTAR ULX program. We also find that NGC 3310 exhibits a factor of ~3-10 elevation of X-ray emission over the other star-forming galaxies. We argue that the excess is most likely explained by the relatively low metallicity of the young stellar population in NGC 3310.
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