Select the best answer for the question.
What's the difference between $126^{1 / 4}$ and $78 \frac{2}{3}$?
A. $58^{5 / 12}$
B. $57^7 / 12$
C. $48^{\frac{1}{3}}$
D. $47^7 / 12$
Answer (1)
Calculate 12 6 1/4 ≈ 3.350368 .
Calculate 78 3 2 = 78.666... .
Calculate the difference: 78.666... − 3.350368 = 75.31629770783216 .
Compare the result with the given options; none are close, indicating a possible error in the question or options.
Explanation
Understanding the Problem We are asked to find the difference between 12 6 1/4 and 78 3 2 , and then choose the closest answer from the given options.
Calculating the Fourth Root of 126 First, let's calculate 12 6 1/4 . This is the fourth root of 126. The result is approximately 3.350368.
Calculating 78 and 2/3 Next, let's calculate 78 3 2 . This is equal to 78 + 3 2 = 78 + 0.666... = 78.666...
Calculating the Difference Now, we find the difference: 78 3 2 − 12 6 1/4 = 78.666... − 3.350368 = 75.31629770783216
Comparing with the Options We need to compare this result to the given options:
A. 5 8 5/12 ≈ 5.429522835002171 B. 5 7 7/12 ≈ 10.574511953557122 C. 4 8 3 1 ≈ 3.634241185664279 D. 4 7 7/12 ≈ 9.449099390116267
None of the options are close to 75.31629770783216. There might be a typo in the question. However, if we assume that the question is asking for 78 3 2 − 12 6 4 1 , then none of the options are correct.
Considering the Reverse Difference However, if the question was intended to ask for 12 6 1/4 − 78 3 2 , then the difference would be 3.350368 − 78.666... = − 75.31629770783216 . In this case, none of the options are close to the absolute value of this result either.
Final Answer and Conclusion Given the available options, it seems there's an error in the problem statement or the provided choices. Since none of the options are close to the calculated difference, we cannot select the best answer from the given choices.
Examples
Understanding the difference between fractional exponents and mixed numbers is crucial in various fields, such as engineering and finance. For example, when calculating the growth rate of an investment or the decay rate of a radioactive substance, you often encounter fractional exponents. Similarly, mixed numbers are used in everyday situations like measuring ingredients for a recipe or determining the amount of material needed for a construction project. Knowing how to accurately compute and compare these values ensures precise calculations and informed decision-making.
