In the following calculations, be sure to use an appropriate number of significant figures. (a) \(3.68 \times 10^{7} \mathrm{g}-4.759 \times 10^{5} \mathrm{g}\) (b) $\frac{6.497 \times 10^{4} \mathrm{m}^{2}}{5.1037 \times 10^{2} \mathrm{m}}$

Given numbers \(3.68 \times 10^{7} \mathrm{g}\) and \(4.759 \times 10^{5} \mathrm{g}\). We need to subtract the two numbers while following the rules of significant figures. Step 1: Write both numbers with the same exponent form. \(3.68 \times 10^{7} \mathrm{g} = 3.680 \times 10^{7} \mathrm{g}\) (4 significant figures) \(4.759 \times 10^{5} \mathrm{g} = 0.04759 \times 10^{7} \mathrm{g}\) (5 significant figures) Step 2: Perform subtraction. \(3.680 \times 10^{7} \mathrm{g} - 0.04759 \times 10^{7} \mathrm{g} = (3.680 - 0.04759) \times 10^{7} \mathrm{g} = 3.63241 \times 10^{7} \mathrm{g}\) Step 3: Round the answer to the lowest number of decimal places in the numbers being subtracted (since we have 3 decimal places in both numbers, no rounding is needed in this case). So, the final answer for part (a) is \(3.63241 \times 10^{7} \mathrm{g}\).

Given numbers \(\frac{6.497 \times 10^{4} \mathrm{m}^{2}}{5.1037 \times 10^{2} \mathrm{m}}\). We need to divide the two numbers while following the rules of significant figures. Step 1: Perform division. \(\frac{6.497 \times 10^{4} \mathrm{m}^{2}}{5.1037 \times 10^{2} \mathrm{m}} = (6.497 \div 5.1037) \times 10^{(4-2)} \mathrm{m} = 1.272594 \times 10^{2} \mathrm{m}\) Step 2: Round the answer to the lowest number of significant figures in the numbers being divided (in this case, 4 significant figures are in both numbers). Final answer for part (b) after rounding: \(1.273 \times 10^{2} \mathrm{m}\). (a) \(3.63241 \times 10^{7} \mathrm{g}\) (b) \(1.273 \times 10^{2} \mathrm{m}\)