Question

Add Numbers in Scientific Notation Practice
A human hair has a diameter of about 4 x 10^-8 inches. A fingernail is about 0.011 inches thicker

Answer 1

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.

1. **Convert 0.011 inches to scientific notation**:
\[
0.011 = 1.1 \times 10^{-2} \text{ inches}
\]

2. **Now we have**:
- Human hair diameter: \(4 \times 10^{-8} \text{ inches}\)
- Fingernail thickness: \(1.1 \times 10^{-2} \text{ inches}\)

3. **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:
\[
1.1 \times 10^{-2} = 1.1 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 1.1 \times 10^{4} \times 10^{-8} = 1.1 \times 10^{4 - 8} = 1.1 \times 10^{-4}
\]

4. **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:
\[
1.1 \times 10^{-4} \text{ dominates the sum, so we can approximate this simply as } 1.1 \times 10^{-4}.
\]

5. **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:
\[
4.0 \times 10^{-4} + 1.1 \times 10^{-4} = (4.0 + 1.1) \times 10^{-4} = 5.1 \times 10^{-4}
\]

6. **Final answer**:
So, the total thickness when you add the diameter of the hair and the thickness of the fingernail in inches is approximately:
\[
5.1 \times 10^{-4} \text{ inches}
\]