To solve the problem [tex]\( \frac{1}{2.67 \times 10^{11}} \)[/tex] and express the result in scientific notation with two decimal places, follow these steps:
1. Understand the Problem:
We need to find the reciprocal of [tex]\( 2.67 \times 10^{11} \)[/tex] and express the result in scientific notation with two decimal places.
2. Calculate the Reciprocal:
The reciprocal of a number [tex]\( x \)[/tex] is [tex]\( \frac{1}{x} \)[/tex]. So we need to find the reciprocal of [tex]\( 2.67 \times 10^{11} \)[/tex].
[tex]\[
\frac{1}{2.67 \times 10^{11}}
\][/tex]
3. Divide the Numbers:
When dividing by a number in scientific notation, we separately consider the coefficient (2.67 in this case) and the power of ten (10[tex]\(^{11}\)[/tex] in this case). The reciprocal of [tex]\( 10^{11} \)[/tex] is [tex]\( 10^{-11} \)[/tex]. So, we find:
[tex]\[
\frac{1}{2.67 \times 10^{11}} = \frac{1}{2.67} \times 10^{-11}
\][/tex]
4. Calculate [tex]\( \frac{1}{2.67} \)[/tex]:
Now we need to perform the division.
[tex]\[
\frac{1}{2.67} \approx 0.3745318352059925
\][/tex]
We can approximate this result to two decimal places.
[tex]\[
0.3745318352059925 \approx 0.37
\][/tex]
5. Combine with Factor of Ten:
Now, recombine this result with the power of ten from step 3:
[tex]\[
0.3745318352059925 \times 10^{-11} \approx 0.37 \times 10^{-11}
\][/tex]
6. Express in Scientific Notation:
Finally, express the result in proper scientific notation. Move the decimal point of 0.37 one place to the right and adjust the exponent accordingly:
[tex]\[
0.37 \times 10^{-11} = 3.7 \times 10^{-12}
\][/tex]
However, we should correct it to two significant figures (as initially planned with the result of two decimal places):
[tex]\[
3.75 \times 10^{-12}
\][/tex]
Therefore, the final result in scientific notation with two decimal places is:
[tex]\[
\boxed{3.75 \times 10^{-12}}
\][/tex]
