To determine the distance from Earth to Proxima Centauri in kilometers, given that the star is 4.2 light years away, we will follow a structured approach to arrive at the solution:
1. Understand the concept of a light year:
A light year is the distance that light travels in one year. This is a significant unit of distance used primarily in astronomy.
2. Convert light years to kilometers:
We know from scientific data that:
[tex]\(1 \text{ light year} = 9.461 \times 10^{12} \text{ km}\)[/tex]
3. Calculate the distance for 4.2 light years:
If 1 light year equals [tex]\(9.461 \times 10^{12}\)[/tex] km, then the distance to Proxima Centauri, which is 4.2 light years away, can be found by multiplying these values:
[tex]\[
\text{Distance} = 4.2 \text{ light years} \times 9.461 \times 10^{12} \text{ km/light year}
\][/tex]
4. Perform the multiplication:
Conducting the multiplication:
[tex]\[
4.2 \times 9.461 = 39.7362
\][/tex]
5. Expressing the answer in standard form:
Since [tex]\(9.461 \times 10^{12}\)[/tex] km is already in standard form, multiplying it by 4.2, we get:
[tex]\[
4.2 \times 9.461 \times 10^{12} = 39.7362 \times 10^{12}
\][/tex]
To express this in proper scientific notation:
[tex]\[
39.7362 \times 10^{12} \text{ km} = 3.97362 \times 10^{13} \text{ km}
\][/tex]
Thus, the distance from Earth to Proxima Centauri is [tex]\(3.97362 \times 10^{13}\)[/tex] kilometers.
