Express in scientific notation. Choose the answer with the proper number of significant figures.
[tex]\begin{array}{l}
\frac{\left(2 \times 10^8\right)\left(6 \times 10^{27}\right)}{12 \times 10^2}= \
10 \times 10^{26} \
020 \times 10^{26} \
2 \times 10^{27}
\end{array}[/tex]
Answer (2)
Calculate the value of the expression: 12 × 1 0 2 ( 2 × 1 0 8 ) ( 6 × 1 0 27 ) = 1 × 1 0 33 .
Consider significant figures: The numbers 2 and 6 have one significant figure, and 12 has two significant figures. The result should have one significant figure.
Express the result in scientific notation with one significant figure: 1 × 1 0 33 .
Compare the result with the available answers and choose the correct one: None of the options are correct, so the final answer is 1 × 1 0 33 .
Explanation
Understanding the Problem We are asked to express the result of the expression 12 × 1 0 2 ( 2 × 1 0 8 ) ( 6 × 1 0 27 ) in scientific notation, choosing the answer with the proper number of significant figures.
Calculating the Value First, let's calculate the numerical value of the expression: 12 × 1 0 2 ( 2 × 1 0 8 ) ( 6 × 1 0 27 ) = 12 2 × 6 × 1 0 2 1 0 8 × 1 0 27 = 12 12 × 1 0 8 + 27 − 2 = 1 × 1 0 33
Considering Significant Figures Now, let's consider significant figures. The numbers given in the problem are: 2 × 1 0 8 , 6 × 1 0 27 , and 12 × 1 0 2 . The numbers 2 and 6 have one significant figure each, and 12 has two significant figures. When multiplying and dividing, the result should have the same number of significant figures as the number with the fewest significant figures. In this case, that's one significant figure.
Comparing with Available Answers The calculated value is 1 × 1 0 33 . Since we need to express the answer with one significant figure, the answer is already in the correct form. Now we compare our result with the available answers:
10 × 1 0 26 can be written as 1 × 1 0 27 . This has one significant figure, but the exponent is incorrect.
0.20 × 1 0 26 can be written as 2.0 × 1 0 25 . This has two significant figures, and the exponent is incorrect.
2 × 1 0 27 . This has one significant figure, but the exponent is incorrect.
Re-evaluating Options However, our calculated result is 1 × 1 0 33 . None of the options match this result. It seems there might be a typo in the provided options. Let's re-evaluate the options:
10 × 1 0 26 = 1 × 1 0 1 × 1 0 26 = 1 × 1 0 27
0.20 × 1 0 26 = 2 × 1 0 − 1 × 1 0 26 = 2 × 1 0 25
2 × 1 0 27
Final Answer Since the correct answer should be 1 × 1 0 33 and none of the options are correct, let's analyze the closest option. If the question was 12 × 1 0 2 ( 2 × 1 0 8 ) ( 6 × 1 0 12 ) , then the answer would be 1 × 1 0 16 . If the question was 12 × 1 0 2 ( 2 × 1 0 8 ) ( 6 × 1 0 17 ) , then the answer would be 1 × 1 0 23 . If the question was 12 × 1 0 2 ( 2 × 1 0 1 ) ( 6 × 1 0 27 ) , then the answer would be 1 × 1 0 26 . The closest answer to 1 × 1 0 33 is none of the options. However, if we consider the first option, 10 × 1 0 26 = 1 × 1 0 27 , it has one significant figure. If the correct answer was 1 × 1 0 27 , then the closest answer would be 10 × 1 0 26 .
Final Conclusion Given the options, the closest answer with one significant figure is 1 × 1 0 27 , which corresponds to 10 × 1 0 26 . However, based on our calculation, the correct answer should be 1 × 1 0 33 . Since none of the options are correct, we will assume there was a typo in the original problem or the options provided. If we assume the answer should have been 1 × 1 0 27 , then the answer would be 10 × 1 0 26 . However, based on the original problem, the correct answer is: 1 × 1 0 33
Examples
Scientific notation is used in many fields, such as physics, astronomy, and engineering, to represent very large or very small numbers. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.017 × 1 0 13 kilometers. Similarly, the size of an atom is approximately 1 × 1 0 − 10 meters. Using scientific notation makes it easier to work with these numbers and perform calculations.
The result of 12 × 1 0 2 ( 2 × 1 0 8 ) ( 6 × 1 0 27 ) is calculated to be 1 × 1 0 33 , which has 1 significant figure. None of the provided options match this answer correctly, but IN terms of significant figures, the closest is 10 × 1 0 26 , equating to 1 × 1 0 27 . Therefore, the accurate calculation leads us to 1 × 1 0 33 .
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