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Sagot :
To determine whether Brian and Joe's answers are correct, let's go through a detailed, step-by-step solution to the problem.
We start with the given fraction:
[tex]\[ \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} \][/tex]
1. Simplify the Powers of 10:
[tex]\[ \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} = \frac{6.89}{7.5} \times 10^{-4 - (-6)} = \frac{6.89}{7.5} \times 10^{2} \][/tex]
2. Calculate the Division of the Coefficients:
[tex]\[ \frac{6.89}{7.5} \approx 0.9186666666666667 \][/tex]
3. Combine the Result of the Division with the Power of 10:
[tex]\[ 0.9186666666666667 \times 10^{2} = 91.86666666666667 \][/tex]
4. Convert to Scientific Notation if Necessary:
[tex]\[ 91.86666666666667 = 9.186666666666667 \times 10^{1} \approx 9.2 \times 10^{1} \][/tex]
However, let's verify the specific steps and logic Brian and Joe used for their calculations.
### Brian's Answer:
Brian claims:
[tex]\[ 92 \times 10^{-10} = \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} = 9.2 \times 10^{-9} \][/tex]
Let's check what Brian actually did:
1. Brian seems to have taken the result [tex]\( 91.86666666666667 \times 10^{0} \)[/tex] and converted it incorrectly to [tex]\( 9.2 \times 10^{1-10} = 9.2 \times 10^{-9} \)[/tex].
Brian was incorrect because the true result is [tex]\( 91.86666666666667 \)[/tex], not [tex]\( 9.2 \times 10^{-9} \)[/tex].
### Joe's Answer:
Joe claims:
[tex]\[ \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} = 0.92 \times 10^2 = 9.2 \times 10^3 \][/tex]
Let's check what Joe actually did:
1. Joe multiplied the result [tex]\( 91.86666666666667 \)[/tex] by [tex]\( 10^{2} \)[/tex], arriving at [tex]\( 91.86666666666667 \times 10^{2} \)[/tex].
Joe converted [tex]\( 0.9186666666666667 \times 10^{2} \)[/tex] incorrectly, arriving at [tex]\( 9.18666666666666667 \times 10^{3} \)[/tex].
### Conclusion:
Both Brian and Joe are incorrect in their final steps:
- Brian’s final answer did not accurately represent the magnitude of the initial fraction and ended up at [tex]\( 9.2 \times 10^{-9} \)[/tex], which is incorrect.
- Joe’s final answer [tex]\( 9.2 \times 10^{3} \)[/tex] stems from incorrect simplification and erroneous steps from his side.
Thus neither Brian nor Joe has given the correct answer.
We start with the given fraction:
[tex]\[ \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} \][/tex]
1. Simplify the Powers of 10:
[tex]\[ \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} = \frac{6.89}{7.5} \times 10^{-4 - (-6)} = \frac{6.89}{7.5} \times 10^{2} \][/tex]
2. Calculate the Division of the Coefficients:
[tex]\[ \frac{6.89}{7.5} \approx 0.9186666666666667 \][/tex]
3. Combine the Result of the Division with the Power of 10:
[tex]\[ 0.9186666666666667 \times 10^{2} = 91.86666666666667 \][/tex]
4. Convert to Scientific Notation if Necessary:
[tex]\[ 91.86666666666667 = 9.186666666666667 \times 10^{1} \approx 9.2 \times 10^{1} \][/tex]
However, let's verify the specific steps and logic Brian and Joe used for their calculations.
### Brian's Answer:
Brian claims:
[tex]\[ 92 \times 10^{-10} = \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} = 9.2 \times 10^{-9} \][/tex]
Let's check what Brian actually did:
1. Brian seems to have taken the result [tex]\( 91.86666666666667 \times 10^{0} \)[/tex] and converted it incorrectly to [tex]\( 9.2 \times 10^{1-10} = 9.2 \times 10^{-9} \)[/tex].
Brian was incorrect because the true result is [tex]\( 91.86666666666667 \)[/tex], not [tex]\( 9.2 \times 10^{-9} \)[/tex].
### Joe's Answer:
Joe claims:
[tex]\[ \frac{6.89 \times 10^{-4}}{7.5 \times 10^{-6}} = 0.92 \times 10^2 = 9.2 \times 10^3 \][/tex]
Let's check what Joe actually did:
1. Joe multiplied the result [tex]\( 91.86666666666667 \)[/tex] by [tex]\( 10^{2} \)[/tex], arriving at [tex]\( 91.86666666666667 \times 10^{2} \)[/tex].
Joe converted [tex]\( 0.9186666666666667 \times 10^{2} \)[/tex] incorrectly, arriving at [tex]\( 9.18666666666666667 \times 10^{3} \)[/tex].
### Conclusion:
Both Brian and Joe are incorrect in their final steps:
- Brian’s final answer did not accurately represent the magnitude of the initial fraction and ended up at [tex]\( 9.2 \times 10^{-9} \)[/tex], which is incorrect.
- Joe’s final answer [tex]\( 9.2 \times 10^{3} \)[/tex] stems from incorrect simplification and erroneous steps from his side.
Thus neither Brian nor Joe has given the correct answer.
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