The danger of radiation from fallout also decreases rapidly with time due in large part to the exponential decay of the individual radionuclides.  A book by Cresson H. Kearny presents data showing that for the first few days after the explosion, the radiation dose rate is reduced by a factor of ten for every seven-fold increase in the number of hours since the explosion.  He presents data showing that "it takes about seven times as long for the dose rate to decay from 1000 roentgens per hour (1000 R/hr) to 10 R/hr (48 hours) as to decay from 1000 R/hr to 100 R/hr (7 hours)."  This is a rule of thumb based on observed data, not a precise relation.