The half-life of radioactive material is the length of time it takes the substance to be halved due to the emission of radiation. If the half-life of a radioactive substance is 50 years, what fraction of the original amount will remain after 150 years?

Given the half-life is 50 years, the original amount will be cut in half three times over a period of 150 years to \(\frac{\mathrm{1} }{\mathrm{2}}\) in 50 years, to \(\frac{\mathrm{1} }{\mathrm{2}} \times \frac{\mathrm{1} }{\mathrm{2}} = \frac{\mathrm{1} }{\mathrm{4}}\) in 100 years, and to \(\frac{\mathrm{1} }{\mathrm{2}} \times \frac{\mathrm{1} }{\mathrm{2}} \times \frac{\mathrm{1} }{\mathrm{2}} = \frac{\mathrm{1} }{\mathrm{8}}\) in 150 years.