To determine how much heavier a grain of rice is than a strand of hair, we need to subtract the mass of the strand of hair from the mass of the grain of rice.
Given:
- Average mass of a strand of hair = \( 2.7 \times 10^{-5} \) ounces
- Average mass of a grain of rice = \( 0.002 \) ounces
First, we convert \( 0.002 \) ounces into scientific notation: \[ 0.002 = 2.0 \times 10^{-3} \text{ ounces} \]
Now, we will subtract the mass of the strand of hair from the mass of the grain of rice: \[ 2.0 \times 10^{-3} - 2.7 \times 10^{-5} \]
To perform this subtraction, we need a common exponent:
Convert \( 2.0 \times 10^{-3} \) to a form that includes \( 10^{-5} \): \[ 2.0 \times 10^{-3} = 2.0 \times 10^{-3} \times \frac{100}{100} = 200 \times 10^{-5} \]
Now we can perform the subtraction: \[ 200 \times 10^{-5} - 2.7 \times 10^{-5} = (200 - 2.7) \times 10^{-5} = 197.3 \times 10^{-5} \]
Next, we express \( 197.3 \) in proper scientific notation: \[ 197.3 \times 10^{-5} = 1.973 \times 10^{-3} \text{ ounces} \quad (\text{moving the decimal point left by 2 places increases the exponent by 2}) \]
Now we express the answer as \( a \times 10^b \):
- \( a = 1.973 \)
- \( b = -3 \)
Thus, \[ a = 1.973, \quad b = -3 \]