A quantitative method to determine the lower energy cutoff (Ec) ofpower-law electron beams is established. We apply this method to the 54 hardX-ray events observed with BATSE/Compton Gamma Ray Observatory (CGRO).The results show that about 75% of the observed broken double power-law spectra of hard X-rays can be explained by a lower energy cutoff in the power-law electronbeams. The values of Ec, varying among the flares, are all greater than the usuallyaccepted 20 keV! On average, Ec is about 69 keV. So high a lower energy cutoff of nonthermal electrons implies that nonthermal electrons might not be as important inpowering solar flares as was previously thought. Further significance of this finding is discussed.