To find the length of a line formed by arranging [tex]$8.4 \times 10^8$[/tex] hydrogen atoms side by side, where each hydrogen atom has a diameter of [tex]$5 \times 10^{-11}$[/tex] meters, we need to multiply these numbers.
1. Express the Step-by-Step Multiplication:
- Diameter of one hydrogen atom: [tex]$5 \times 10^{-11}$[/tex] meters
- Number of hydrogen atoms: [tex]$8.4 \times 10^8$[/tex]
2. Perform the Multiplication:
[tex]\[
(5 \times 10^{-11}) \times (8.4 \times 10^8) = (5 \times 8.4) \times (10^{-11} \times 10^8)
\][/tex]
3. Multiply the Coefficients:
Calculate [tex]\(5 \times 8.4\)[/tex]:
[tex]\[
5 \times 8.4 = 42
\][/tex]
4. Multiply the Powers of 10:
Calculate [tex]\(10^{-11} \times 10^8\)[/tex]:
[tex]\[
10^{-11} \times 10^8 = 10^{-11 + 8} = 10^{-3}
\][/tex]
5. Combine the Results:
Combine the products to get:
[tex]\[
42 \times 10^{-3}
\][/tex]
6. Express in Standard Scientific Notation:
The result [tex]\(42 \times 10^{-3}\)[/tex] meters can be converted to standard scientific notation. To do this, we ensure the coefficient is between 1 and 10:
[tex]\[
42 \times 10^{-3} = 4.2 \times 10^{-2}
\][/tex]
Therefore, the length of the line of hydrogen atoms can be expressed in scientific notation as [tex]\(4.2 \times 10^{-2}\)[/tex] meters.
Additionally, in different forms, the length can also be expressed as:
- [tex]\(0.042\)[/tex] meters, which is the same as [tex]\(4.2 \times 10^{-2}\)[/tex] meters.
- Another possible form, although less common for this value, is [tex]\(4.2 \times 10^{-3}\)[/tex] meters, but as per the logical derivation, the most accurate scientific notation is [tex]\(4.2 \times 10^{-2}\)[/tex] meters.
In conclusion, the correct length in scientific notation is:
[tex]\[ \boxed{4.2 \times 10^{-2} \text{ meters}} \][/tex]
This matches the provided solution and confirms our step-by-step derivation.
