To determine the best scientific notation for the given standard notation [tex]\( 6,840,000,000 \, \text{m} \)[/tex], we need to follow these steps:
1. Identify the given number in standard notation:
[tex]\[
6,840,000,000 \, \text{m}
\][/tex]
2. Convert the number to scientific notation:
Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. The format is:
[tex]\[
a \times 10^b
\][/tex]
where [tex]\( 1 \leq a < 10 \)[/tex] and [tex]\( b \)[/tex] is an integer.
3. Place the decimal point:
For [tex]\( 6,840,000,000 \)[/tex], we move the decimal point 9 places to the left to get:
[tex]\[
6.84
\][/tex]
This gives us:
[tex]\[
6.84 \times 10^9
\][/tex]
4. Check the precision:
According to typical scientific notation rules, we often round to two significant digits if needed. In this case, [tex]\( 6.840000000 \)[/tex] indicates we keep three significant digits:
[tex]\[
6.84 \times 10^9
\][/tex]
5. Compare with the given choices:
- [tex]\( 68.4 \times 10^8 \, \text{m} \)[/tex]: This is not in true scientific notation form because [tex]\( 68.4 \)[/tex] is greater than [tex]\( 10 \)[/tex].
- [tex]\( 6.84 \times 10^8 \, \text{m} \)[/tex]: This does not represent the initial number correctly because it represents a value of [tex]\( 684,000,000 \)[/tex], much smaller than [tex]\( 6,840,000,000 \)[/tex].
- [tex]\( 6.84 \times 10^{-9} \, \text{m} \)[/tex]: This represents an extremely small number, not the large value we have.
- [tex]\( 6.84 \times 10^9 \, \text{m} \)[/tex]: This correctly represents [tex]\( 6,840,000,000 \, \text{m} \)[/tex].
Thus, the best representation in scientific notation for the standard notation [tex]\( 6,840,000,000 \, \text{m} \)[/tex] is:
[tex]\[
6.84 \times 10^9 \, \text{m}
\][/tex]
