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.

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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:**
  $$
  1 \, \text{cm} = 10^{-2} \, \text{meters}
  $$

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**Summary of Answers:**
  
**a.** $$ 10^{-6} $$ meters  
**b.** $$ 10^{-15} $$ meters  
**c.** $$ 10^{-2} $$ meters
1 micrometer = 0.000001 meters 1 femtometer = 0.000000000000001 meters 1 centimeter = 0.01 meters

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