Simplify: [tex]$27_6-5^2 / 3+4^3 / 4$[/tex]
A. [tex]$1 / 4$[/tex]
B. [tex]$1 %$[/tex]
C. [tex]$1 \frac{1}{4} / 4$[/tex]
D. [tex]$1^3 / 4$[/tex]
Answer (2)
Convert 2 7 6 to base 10: 2 7 6 = 19 .
Calculate 5 2 = 25 and 4 3 = 64 .
Evaluate the expression: 19 − 25/3 + 64/4 = 19 − 8 3 1 + 16 = 26 3 2 .
Compare the result with the given options; none match, indicating a possible error in the question or options.
Explanation
Understanding the Problem We need to simplify the expression 2 7 6 − 5 2 /3 + 4 3 /4 and choose the correct option from the given choices.
Converting to Base 10 First, we convert 2 7 6 to base 10. 2 7 6 = 2 × 6 1 + 7 × 6 0 = 2 × 6 + 7 × 1 = 12 + 7 = 19 .
Calculating Powers Next, we calculate 5 2 = 5 × 5 = 25 and 4 3 = 4 × 4 × 4 = 64 .
Performing Division Now, we divide 5 2 by 3: 25/3 = 3 25 = 8 3 1 .
Performing Division And we divide 4 3 by 4: 64/4 = 16 .
Substituting and Simplifying Now we substitute these values back into the original expression: 19 − 3 25 + 16 = 19 − 8 3 1 + 16 = 35 − 8 3 1 = 26 3 2 .
Converting to Decimal We need to express 26 3 2 as a decimal to compare with the options. 26 3 2 = 26.666...
Comparing with Options Now let's examine the options: A. 1/4 = 0.25 B. 1% = 0.01 C. 1 4 1 /4 = 4 5 /4 = 16 5 = 0.3125 D. 1 3 /4 = 1/4 = 0.25 None of the options match our result of 26 3 2 . There seems to be an error in the problem statement or the options provided. However, if we assume the question meant ( 2 7 6 − 5 2 ) /3 + 4 3 /4 , then we have ( 19 − 25 ) /3 + 64/4 = − 6/3 + 16 = − 2 + 16 = 14 . Still, none of the options match. Let's assume the question meant 2 7 6 − ( 5 2 /3 + 4 3 ) /4 , then we have 19 − ( 25/3 + 64 ) /4 = 19 − ( 25/3 + 192/3 ) /4 = 19 − ( 217/3 ) /4 = 19 − 217/12 = ( 228 − 217 ) /12 = 11/12 = 0.9166... Still, none of the options match.
Final Evaluation and Conclusion Let's re-evaluate the expression 2 7 6 − 5 2 /3 + 4 3 /4 as 19 − 25/3 + 64/4 = 19 − 25/3 + 16 = 35 − 25/3 = ( 105 − 25 ) /3 = 80/3 = 26 3 2 . Since none of the options match, it is possible that there is a typo in the question or the options. However, if we are forced to choose the closest option, we can consider option C, 1 4 1 /4 = 0.3125 , which is far from the correct answer. Therefore, we can conclude that there is an error in the question or the options.
Examples
When simplifying mathematical expressions, the order of operations (PEMDAS/BODMAS) is crucial. This problem reinforces the importance of converting numbers to a common base, performing exponents, and following the correct order of division, multiplication, addition, and subtraction. Understanding these concepts is essential in various fields, such as engineering, physics, and computer science, where complex calculations are frequently performed. For example, calculating electrical circuit parameters or analyzing data sets requires a solid grasp of mathematical simplification.
The expression 2 7 6 − 3 5 2 + 4 4 3 simplifies to 3 80 or approximately 26 3 2 . None of the given answer options match this result, indicating a possible error in the question. As it stands, I cannot select any of the provided options as the correct answer.
;
