Sure! Let's go step-by-step through the problem:
### Part (a)
We need to write the distance of one light year in standard form. The distance that light travels in one year is given as [tex]\( 9,463,700,000,000 \)[/tex] km.
To express this in standard form, we need to move the decimal point to the rightmost position just after the first digit and count the number of places the decimal point is moved. This gives us:
[tex]\[ 9463700000000 \text{ km} = 9.4637 \times 10^{12} \text{ km} \][/tex]
Rounding to one decimal place, this becomes:
[tex]\[ 9.5 \times 10^{12} \text{ km} \][/tex]
So, in standard form, one light year is:
[tex]\[ \boxed{9.5 \times 10^{12} \text{ km}} \][/tex]
### Part (b)
The distance from the Sun to Earth is given as [tex]\( 0.000016 \)[/tex] light years. We need to write this in standard form.
To do this, move the decimal point 5 places to the right:
[tex]\[ 0.000016 = 1.6 \times 10^{-5} \][/tex]
So, the distance from the Sun to Earth in light years in standard form is:
[tex]\[ \boxed{1.6 \times 10^{-5} \text{ light years}} \][/tex]
### Part (c)
Proxima Centauri is 4.2 light years away from Earth. We need to convert this distance into kilometers and then express it in standard form.
Given that one light year is [tex]\( 9.5 \times 10^{12} \)[/tex] km from part (a), we calculate the distance to Proxima Centauri as follows:
[tex]\[ \text{Distance in km} = 4.2 \text{ light years} \times 9.4637 \times 10^{12} \text{ km/light year} \][/tex]
Calculating this gives:
[tex]\[ 4.2 \times 9.4637 \times 10^{12} \approx 39.74754 \times 10^{12} \text{ km} \][/tex]
Expressing [tex]\( 39.74754 \times 10^{12} \)[/tex] in standard form, and rounding to one decimal place gives us:
[tex]\[ 4.0 \times 10^{13} \text{ km} \][/tex]
Thus, the distance to Proxima Centauri in standard form is:
[tex]\[ \boxed{4.0 \times 10^{13} \text{ km}} \][/tex]
The exact value without rounding is:
[tex]\[ \boxed{39747540000000.0 \text{ km}} \][/tex]
These are the step-by-step solutions for parts (a), (b), and (c) of the question.
