A quantitative method to determine the lower energy cutoff (E,) of power-la
w electron beams is established. We apply this method to the 54 hard X-ray
events observed with BATSE/Compton Gamma Ray Observatory (CGRO). The result
s show that about 75% of the observed broken double power-law spectra of ha
rd X-rays can be explained by a lower energy cutoff in the power-law electr
on beams. The values of E,, varying among the flares, are all greater than
the usually accepted 20 keV! On average, E-c is about 69 keV. So high a low
er energy cutoff of nonthermal electrons implies that nonthermal electrons
might not be as important in powering solar flares as was previously though
t. Further significance of this finding is discussed.