27. Metric System The metric system has names for very small lengths. a. One micrometer is \( 10^{3} \) times the length of one nanometer. One nanometer is \( 10^{-9} \) meter. Write one micrometer in meters. b. One femtometer is \( 10^{3} \) times the length of one attometer. One attometer is \( 10^{-18} \) meter. Write one femtometer in meters. c. One centimeter is \( 10^{10} \) times the length of one picometer. One picometer is \( 10^{-12} \) meter. Write one centimeter in meters.

Let's solve each part step by step using the relationships between metric prefixes and their corresponding powers of ten. ### a. Micrometer in Meters - **Given:** - 1 nanometer (nm) = \( 10^{-9} \) meters - 1 micrometer (µm) = \( 10^{3} \) nanometers - **Calculation:** \[ 1 \, \mu\text{m} = 10^{3} \times 1 \, \text{nm} = 10^{3} \times 10^{-9} \, \text{m} = 10^{-6} \, \text{meters} \] - **Answer:** \[ 1 \, \mu\text{m} = 10^{-6} \, \text{meters} \] ### b. Femtometer in Meters - **Given:** - 1 attometer (am) = \( 10^{-18} \) meters - 1 femtometer (fm) = \( 10^{3} \) attometers - **Calculation:** \[ 1 \, \text{fm} = 10^{3} \times 1 \, \text{am} = 10^{3} \times 10^{-18} \, \text{m} = 10^{-15} \, \text{meters} \] - **Answer:** \[ 1 \, \text{fm} = 10^{-15} \, \text{meters} \] ### c. Centimeter in Meters - **Given:** - 1 picometer (pm) = \( 10^{-12} \) meters - 1 centimeter (cm) = \( 10^{10} \) picometers - **Calculation:** \[ 1 \, \text{cm} = 10^{10} \times 1 \, \text{pm} = 10^{10} \times 10^{-12} \, \text{m} = 10^{-2} \, \text{meters} \] Alternatively, it's widely known that: \[ 1 \, \text{cm} = 0.01 \, \text{meters} = 10^{-2} \, \text{meters} \] - **Answer:** \[ 1 \, \text{cm} = 10^{-2} \, \text{meters} \] --- **Summary of Answers:** **a.** \( 10^{-6} \) meters **b.** \( 10^{-15} \) meters **c.** \( 10^{-2} \) meters