The radioactive substance cesium- 137 has a half-life of 30 years. The amount \( A(t) \) (in grams) of a sample of cesium- 137 remaining after \( t \) years is given by the following exponential function. \( A(t)=266\left(\frac{1}{2}\right)^{\frac{t}{30}} \) Find the amount of the sample remaining after 20 years and after 100 years. Round your answers to the nearest gram as necessary. Amount after 20 years: Amount after 100 years: Dgrams

Calculate or simplify the expression \( 266\left(\frac{1}{2}\right)^{\frac{20}{30}} \). Calculate the value by following steps: - step0: Calculate: \(266\left(\frac{1}{2}\right)^{\frac{20}{30}}\) - step1: Reduce the fraction: \(266\left(\frac{1}{2}\right)^{\frac{2}{3}}\) - step2: Evaluate the power: \(266\times \frac{\sqrt[3]{2}}{2}\) - step3: Multiply: \(133\sqrt[3]{2}\) Calculate or simplify the expression \( 266\left(\frac{1}{2}\right)^{\frac{100}{30}} \). Calculate the value by following steps: - step0: Calculate: \(266\left(\frac{1}{2}\right)^{\frac{100}{30}}\) - step1: Reduce the fraction: \(266\left(\frac{1}{2}\right)^{\frac{10}{3}}\) - step2: Evaluate the power: \(266\times \frac{\sqrt[3]{4}}{16}\) - step3: Multiply: \(\frac{133\sqrt[3]{4}}{8}\) The amount of the sample remaining after 20 years is approximately 167.57 grams, and after 100 years is approximately 26.39 grams.