To convert the thickness of a piece of paper, which is 0.097 millimeters, into scientific notation, let's break it down step-by-step.
1. Determine the Leading Digit:
Identify the most significant digit in the value. For 0.097, the leading digit is 9.7 when we move the decimal point two places to the right.
2. Establish the Exponent:
Since we moved the decimal point two places to the right to get from 0.097 to 9.7, we need to account for this shift by using a factor of [tex]\(10^{-2}\)[/tex]. This is because moving the decimal point to the right by two positions is equivalent to multiplying by [tex]\(10^{-2}\)[/tex] (which is the same as dividing by 100).
3. Combine the Parts:
The scientific notation is therefore formed by combining the leading digit and the appropriate power of 10:
[tex]\[
0.097 = 9.7 \times 10^{-2}
\][/tex]
Let's also review other possible forms given in the options to ensure the accuracy of our scientific notation:
- [tex]\(9.7 \times 10^2\)[/tex]: This would mean 970, which is not close to 0.097.
- [tex]\(97 \times 10^{-3}\)[/tex]: Although [tex]\(97 \times 10^{-3}\)[/tex] simplifies to 0.097, it's not in the standardized form for scientific notation (which expects a single non-zero digit before the decimal point).
- [tex]\(9.7 \times 10^3\)[/tex]: This would mean 9700, which is significantly different from 0.097.
Therefore, the correct scientific notation for 0.097 millimeters is:
[tex]\[
9.7 \times 10^{-2}
\][/tex]
So, the answer is:
[tex]\[
9.7 \times 10^{-2}
\][/tex]
