To add the diameter of a human hair (4 x 10^-8 inches) to the thickness of a fingernail (0.011 inches), we first need to express both quantities in the same format. We'll convert 0.011 inches into scientific notation.
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Convert 0.011 inches to scientific notation: \[ 0.011 = 1.1 \times 10^{-2} \text{ inches} \]
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Now we have:
- Human hair diameter: \(4 \times 10^{-8} \text{ inches}\)
- Fingernail thickness: \(1.1 \times 10^{-2} \text{ inches}\)
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Align the powers of 10: To add these numbers, we need to express them with the same exponent. The smaller exponent here is \(10^{-8}\). We can express \(1.1 \times 10^{-2}\) with this exponent: \[ 1.1 \times 10^{-2} = 1.1 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 1.1 \times 10^{4} \times 10^{-8} = 1.1 \times 10^{4 - 8} = 1.1 \times 10^{-4} \]
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Now we can add them together: \[ 4 \times 10^{-8} + 1.1 \times 10^{-4} \] Notice that \(10^{-4}\) is much larger than \(10^{-8}\), so we can say: \[ 1.1 \times 10^{-4} \text{ dominates the sum, so we can approximate this simply as } 1.1 \times 10^{-4}. \]
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More precisely, to express it with the same exponent: We need to convert \(4 \times 10^{-8}\) to \(10^{-4}\): \[ 4 \times 10^{-8} = 0.00004 = 0.00004 \times \frac{10^{6}}{10^{6}} = 4.0 \times 10^{-2} \times 10^{-8} = 0.00004 \times 10^{-4} \] So, \[ 4 \times 10^{-8} \text{ in terms of } 10^{-4} is \text{approximately } 0.0000004 \times 10^{-4} \]
Thus, we can perform the addition: \[ 4.0 \times 10^{-4} + 1.1 \times 10^{-4} = (4.0 + 1.1) \times 10^{-4} = 5.1 \times 10^{-4} \]
- Final answer: So, the total thickness when you add the diameter of the hair and the thickness of the fingernail in inches is approximately: \[ 5.1 \times 10^{-4} \text{ inches} \]
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