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<|im_start|>system
Your role as an assistant involves reconstructing the internal reasoning process that connects a provided question to its correct answer. Your task is to methodically reverse-engineer the logical steps, demonstrating a full cycle of analysis, summarization, idea generation, verification, error correction, and iterative refinement. Please structure your response into two distinct parts: Solution and Thought. In the Solution section, present the given correct answer in a precise and clear format: <|begin_of_solution|> {provided correct solution} <|end_of_solution|>. In the Thought section, articulate the step-by-step cognitive journey that leads to the solution. Use the specified format: <|begin_of_thought|> {detailed thought process with steps separated by '\n\n'} <|end_of_thought|>. Each step should reflect analytical breakdowns, synthesis of key points, generation of logical pathways, validation of each step's accuracy, refinement of any missteps, and reassessment of previous conclusions. The focus is solely on depicting the internal, structured thinking that arrives at the provided solution.<|im_end|>
<|im_start|>user
Return your final response within \boxed{}. The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$<|im_end|>
<|im_start|>assistant
<|begin_of_solution|>
To determine the value of \([(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]\) where the operation \(\otimes\) is defined by \(a \otimes b = \frac{a^2}{b}\), we proceed as follows:
First, compute \(1 \otimes 2\):
\[
1 \otimes 2 = \frac{1^2}{2} = \frac{1}{2}
\]
Next, use this result to compute \((1 \otimes 2) \otimes 3\):
\[
\left(\frac{1}{2}\right) \otimes 3 = \frac{\left(\frac{1}{2}\right)^2}{3} = \frac{\frac{1}{4}}{3} = \frac{1}{12}
\]
Now, compute \(2 \otimes 3\):
\[
2 \otimes 3 = \frac{2^2}{3} = \frac{4}{3}
\]
Then, use this result to compute \(1 \otimes (2 \otimes 3)\):
\[
1 \otimes \left(\frac{4}{3}\right) = \frac{1^2}{\frac{4}{3}} = \frac{1}{\frac{4}{3}} = \frac{3}{4}
\]
Finally, find the difference between the two results:
\[
\frac{1}{12} - \frac{3}{4} = \frac{1}{12} - \frac{9}{12} = \frac{1 - 9}{12} = \frac{-8}{12} = -\frac{2}{3}
\]
Thus, the answer is \(\boxed{A}\).
<|end_of_solution|>
<|begin_of_thought|>
|
Okay, let me try to figure out this problem. So, we have this operation defined as a⊗b = a²/b. And we need to compute [(1⊗2)⊗3] - [1⊗(2⊗3)]. Then choose the correct answer from the options given. Alright, let's break it down step by step.
First, I need to remember that the operation ⊗ is not associative, right? Because the problem is asking for the difference between two different groupings: (1⊗2)⊗3 and 1⊗(2⊗3). So, the order in which we perform the operations matters here. That's probably why there's a subtraction between them.
Let me start by computing each part separately. Let's tackle the first part: (1⊗2)⊗3.
Starting with the innermost operation, which is 1⊗2. According to the definition, a⊗b = a²/b. So here, a is 1 and b is 2. Plugging those in: 1² / 2 = 1/2. So, 1⊗2 equals 1/2.
Now, we take that result and perform the next operation with 3. So, (1⊗2)⊗3 becomes (1/2)⊗3. Again, using the same definition: a is now 1/2 and b is 3. So, ( (1/2)² ) / 3 = (1/4) / 3 = 1/12. So, (1⊗2)⊗3 equals 1/12.
Alright, that's the first part. Now let's compute the second part: 1⊗(2⊗3). Again, starting with the innermost operation, which is 2⊗3. Applying the definition: a is 2 and b is 3. So, 2² / 3 = 4/3. Therefore, 2⊗3 equals 4/3.
Now, we need to compute 1⊗(4/3). Here, a is 1 and b is 4/3. Using the operation definition: 1² / (4/3) = 1 / (4/3) = 3/4. So, 1⊗(2⊗3) equals 3/4.
Now, the problem asks for the difference between the two results: [(1⊗2)⊗3] - [1⊗(2⊗3)] = (1/12) - (3/4). To subtract these fractions, they need a common denominator. The denominators are 12 and 4, so 12 is the common denominator.
Converting 3/4 to twelfths: 3/4 = 9/12. So, 1/12 - 9/12 = (1 - 9)/12 = -8/12. Simplifying that fraction by dividing numerator and denominator by 4: -8/12 = -2/3.
Hmm, looking at the answer choices, option A is -2/3. So, is that the answer? Wait, but let me double-check my calculations to make sure I didn't make a mistake somewhere.
First, checking (1⊗2): 1² / 2 = 1/2. Correct. Then, (1/2)⊗3: (1/2)² / 3 = (1/4)/3 = 1/12. That seems right.
Now, for 2⊗3: 2² / 3 = 4/3. Correct. Then, 1⊗(4/3): 1² / (4/3) = 1 / (4/3) = 3/4. Yes, that's correct.
Subtracting 3/4 from 1/12: 1/12 - 3/4. Convert 3/4 to 9/12, so 1/12 - 9/12 = -8/12 = -2/3. Yes, that all checks out. So the answer should be -2/3, which is option A.
Wait, but let me think again. The operation is defined for all nonzero numbers, so we don't have any issues with division by zero here. 2⊗3 is 4/3, which is fine, and then 1⊗(4/3) is 3/4. Correct.
Alternatively, maybe there's a different way to approach the problem? Let me try expanding both expressions using variables to see if there's a pattern.
Let’s denote the first expression: (a⊗b)⊗c. Using the definition:
First, compute a⊗b = a²/b.
Then, take that result and ⊗ with c: (a²/b)⊗c = ( (a²/b)² ) / c = a⁴ / (b² c).
Now, the second expression: a⊗(b⊗c). First compute b⊗c = b²/c.
Then, a⊗(b²/c) = a² / (b²/c) = a² * (c / b²) = (a² c) / b².
Therefore, the difference between the two expressions is:
(a⁴ / (b² c)) - (a² c / b²) = (a⁴ - a² c²) / (b² c) = a² (a² - c²) / (b² c).
Hmm, factoring that, it's a² (a - c)(a + c) / (b² c).
But in our specific problem, a = 1, b = 2, c = 3. Plugging those values in:
1² (1 - 3)(1 + 3) / (2² * 3) = 1 * (-2)(4) / (4 * 3) = (-8) / 12 = -2/3. Same result. So that confirms the answer is indeed -2/3.
Therefore, I think my initial calculation was correct, and the answer is option A.
**Final Answer**
\boxed{A}
<|end_of_thought|><|im_end|>
|
Okay, let me try to figure out this problem. So, we have this operation defined as a⊗b = a²/b. And we need to compute [(1⊗2)⊗3] - [1⊗(2⊗3)]. Then choose the correct answer from the options given. Alright, let's break it down step by step.
First, let me understand the operation. The tensor product here, ⊗, takes two nonzero numbers a and b and returns a squared divided by b. So, it's a binary operation that's a bit different from the usual addition or multiplication. Importance to remember that order matters here because the denominator is the second operand. So, a⊗b is not the same as b⊗a, right? For example, if a=2 and b=3, then 2⊗3 would be 4/3, but 3⊗2 would be 9/2, which is 4.5. So definitely not commutative.
Now, the expression we need to compute is [(1⊗2)⊗3] minus [1⊗(2⊗3)]. This looks like it's testing the associativity of the operation ⊗. Since the question is asking for the difference between the two expressions, if the operation were associative, the result would be zero. But since it's not necessarily associative, we might get a non-zero value. The answer choices include both positive and negative fractions, so we need to calculate carefully.
Let me first compute (1⊗2)⊗3. Let's do this step by step. First, compute 1⊗2. According to the definition, a=1 and b=2. So, 1⊗2 = (1)² / 2 = 1/2. Then, take this result and tensor it with 3. So, (1⊗2)⊗3 becomes (1/2)⊗3. Applying the operation again: a is now 1/2 and b is 3. So, (1/2)² / 3 = (1/4) / 3 = 1/12. So, [(1⊗2)⊗3] = 1/12.
Now, let's compute the other part: 1⊗(2⊗3). Again, starting with the innermost operation. First, compute 2⊗3. Here, a=2 and b=3, so 2² / 3 = 4/3. Then, tensor this result with 1. So, 1⊗(4/3). Here, a=1 and b=4/3. Therefore, (1)² / (4/3) = 1 / (4/3) = 3/4. So, [1⊗(2⊗3)] = 3/4.
Now, subtract the second result from the first: (1/12) - (3/4). To subtract these fractions, they need a common denominator. The denominators are 12 and 4. 12 is a multiple of 4, so converting 3/4 to twelfths: 3/4 = 9/12. Therefore, 1/12 - 9/12 = (1 - 9)/12 = -8/12. Simplifying that, divide numerator and denominator by 4: -2/3. Hmm, so the result is -2/3. Let me check the answer choices. Option A is -2/3. So, is that the answer? Wait, let me verify my calculations again to make sure I didn't make a mistake.
First part: (1⊗2)⊗3. 1⊗2 is 1²/2 = 1/2. Then, (1/2)⊗3 is (1/2)² / 3 = (1/4)/3 = 1/12. That seems right. Then, 2⊗3 is 4/3. Then, 1⊗(4/3) is 1²/(4/3) = 1/(4/3) = 3/4. Then, 1/12 - 3/4. Convert 3/4 to 9/12, so 1/12 - 9/12 = -8/12 = -2/3. Yes, that's correct. So, answer A.
Wait, but just to be thorough, let me check if I applied the operation correctly. The operation is a⊗b = a²/b. So, when we have (1⊗2)⊗3, the first operation is 1⊗2, which is 1²/2 = 1/2. Then, the next operation is (1/2)⊗3. Here, a is 1/2 and b is 3, so (1/2)² / 3 = (1/4)/3 = 1/12. Correct.
For the second part, 1⊗(2⊗3). First, 2⊗3 is 4/3. Then, 1⊗(4/3) is 1² / (4/3) = 1 * (3/4) = 3/4. So, 3/4. Then subtract: 1/12 - 3/4. 3/4 is 9/12, so 1/12 - 9/12 = -8/12 = -2/3. That all checks out.
Alternatively, maybe I can do the calculations using fractions step by step to avoid confusion. Let me write each step again:
First expression: (1⊗2)⊗3
Step 1: Compute 1⊗2
a = 1, b = 2
= (1)^2 / 2 = 1/2
Step 2: Take result from Step 1 and tensor with 3
a = 1/2, b = 3
= (1/2)^2 / 3 = (1/4) / 3 = 1/12
Second expression: 1⊗(2⊗3)
Step 1: Compute 2⊗3
a = 2, b = 3
= (2)^2 / 3 = 4/3
Step 2: Take result from Step 1 and tensor with 1
a = 4/3, b = 1
= (4/3)^2 / 1 = (16/9) / 1 = 16/9
Wait, hold on! Wait a second. Wait, no. Wait, in the operation 1⊗(2⊗3), the first operand is 1 and the second operand is (2⊗3). So, when we compute 1⊗(4/3), it's a=1 and b=4/3. Therefore, (1)^2 / (4/3) = 1 / (4/3) = 3/4, not (4/3)^2. Wait, that's a key point here. I think I made a mistake here.
Hold on, let me re-examine that step. So in the second expression, 1⊗(2⊗3). First, compute 2⊗3, which is 4/3. Then, compute 1⊗(4/3). Here, a is 1 and b is 4/3. Therefore, 1² divided by (4/3) is 1/(4/3) = 3/4. But in my previous calculation, I thought of a=4/3, but no, the first operand is 1. So, the a in the operation is 1, and the b is 4/3. Therefore, it's 1² divided by 4/3, which is 1*(3/4) = 3/4. So that part is correct. So, 1⊗(4/3) is 3/4. So, the second expression is 3/4. Then subtract: 1/12 - 3/4. Which is indeed -2/3. So, that part is correct. Wait, but in my initial calculation, I had that. So, maybe my mistake was in writing here, but actually, no. So, perhaps I was confused, but the calculation is correct.
Alternatively, maybe I need to check once again. Let's write all steps again:
First part:
(1⊗2)⊗3:
1⊗2 = 1² / 2 = 1/2
Then, (1/2)⊗3 = (1/2)² / 3 = (1/4)/3 = 1/12
Second part:
1⊗(2⊗3):
2⊗3 = 2² / 3 = 4/3
Then, 1⊗(4/3) = 1² / (4/3) = 1 / (4/3) = 3/4
Subtraction:
1/12 - 3/4 = 1/12 - 9/12 = -8/12 = -2/3
Yes, that's correct. So answer is A, -2/3.
But let me check once again because sometimes operations like this can be tricky with the order. Let me verify each step once more.
First, [(1⊗2)⊗3]:
1⊗2: a=1, b=2. So, 1² / 2 = 1/2. Correct.
Then, (1/2)⊗3: a=1/2, b=3. So, (1/2)^2 = 1/4. Divide by 3: 1/4 ÷ 3 = 1/12. Correct.
Now, [1⊗(2⊗3)]:
2⊗3: a=2, b=3. 2² /3 = 4/3. Correct.
Then, 1⊗(4/3): a=1, b=4/3. 1² / (4/3) = 1 * 3/4 = 3/4. Correct.
Subtract: 1/12 - 3/4. Convert to twelfths: 3/4 = 9/12. 1/12 - 9/12 = -8/12 = -2/3. Correct.
Therefore, the answer is A, -2/3. So, I think that's right. Let me check the options again: (A) -2/3, (B) -1/4, (C) 0, (D) 1/4, (E) 2/3. So, yes, A is the correct answer.
But just to ensure, let me think if there's any alternative way to approach this problem. Maybe by generalizing the operation.
The operation ⊗ is defined as a⊗b = a² / b. Let's see if we can find a pattern or a formula for expressions like (a⊗b)⊗c - a⊗(b⊗c).
Let me compute (a⊗b)⊗c:
First, a⊗b = a² / b. Then, (a² / b)⊗c = (a² / b)² / c = a⁴ / (b² c)
Now, a⊗(b⊗c):
First, b⊗c = b² / c. Then, a⊗(b² / c) = a² / (b² / c) = a² * c / b².
Therefore, the difference is:
(a⁴ / (b² c)) - (a² c / b²) = (a² / b²) (a² / c - c)
Factor out (a² / b²):
(a² / b²)(a² / c - c) = (a² / b²)( (a² - c²) / c )
Wait, maybe simplifying this expression:
= (a² (a² - c²)) / (b² c )
But perhaps plugging in specific values for a, b, c. Let's take a=1, b=2, c=3.
Then, the difference would be:
[(1)^4 / ((2)^2 * 3)] - [ (1)^2 * 3 / (2)^2 ] = (1 / (4 * 3)) - (3 / 4) = (1/12) - (9/12) = -8/12 = -2/3. Which matches our previous calculation.
Alternatively, using the general formula:
(1² / 2²)( (1² - 3²)/3 ) = (1/4)( (1 - 9)/3 ) = (1/4)(-8/3) = -8/12 = -2/3. Same result.
Therefore, the answer is indeed -2/3, which is option A.
I think this confirms that the answer is correct. I don't see any mistakes in the calculations now. So, the final answer is A.
**Final Answer**
\boxed{A}
| 0
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] |
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let me try to figure out this problem. So Doug has a square window made up of 8 equal-sized p(...TRUNCATED)
| "Okay, let me try to figure out this problem. So Doug has a square window made up of 8 equal-sized p(...TRUNCATED)
| 1
| [-0.05507265403866768,-0.000021219027985353023,-0.34866154193878174,-1.8048176765441895,-0.001975015(...TRUNCATED)
| [-3.576278118089249e-7,-3.4570634852570947e-6,0.0,0.0,-2.3841855067985307e-7,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let's see. I need to find the value of n given this polynomial P(x) of degree 3n that satisfi(...TRUNCATED)
| "Okay, let me try to work through this problem step by step. So, we have a polynomial P(x) of degree(...TRUNCATED)
| 2
| [-0.12732812762260437,-0.00002288792165927589,-0.7581201195716858,-1.8047798871994019,-0.00229750201(...TRUNCATED)
| [-0.000058410845667822286,-1.1920928244535389e-7,0.0,-1.0728830375228426e-6,-0.000010132738680113107(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. The function is given as f(x) = a x² - (...TRUNCATED)
| "Okay, let me try to solve this problem step by step. The function is given as f(x) = a x² - √2, (...TRUNCATED)
| 3
| [-0.07908769696950912,-0.000027894584491150454,-0.38700801134109497,-1.6004337072372437,-0.004084103(...TRUNCATED)
| [-5.960462772236497e-7,-2.3841855067985307e-7,0.0,0.0,-8.344646857949556e-7,0.0,0.0,-1.1920928244535(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let's try to figure out this problem step by step. So, the question is about comparing the to(...TRUNCATED)
| "Okay, let's see. The problem is about comparing the total cost of buying an appliance in-store vers(...TRUNCATED)
| 4
| [-0.03006083145737648,-0.000023364747903542593,-0.3134635388851166,-0.31329232454299927,-0.552381634(...TRUNCATED)
| [-4.768370445162873e-7,-3.576278118089249e-7,0.0,-1.1920928244535389e-7,-5.960462772236497e-7,0.0,0.(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let's try to tackle this geometry problem step by step. First, I need to visualize the setup.(...TRUNCATED)
| "Okay, let's try to tackle this geometry problem. Hmm, so we have points A, B, C, D, E, and F lying (...TRUNCATED)
| 5
| [-0.07933791726827621,-0.000027656173188006505,-0.3872584402561188,-0.3869107663631439,-0.3539535999(...TRUNCATED)
| [-5.960462772236497e-7,-2.3841855067985307e-7,0.0,0.0,-7.152555099310121e-7,0.0,-1.1920928244535389e(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let's see. The problem is about Judy's hits during the softball season. She had a total of 35(...TRUNCATED)
| "Okay, let's see. The problem is about Judy's hits during the softball season. She had a total of 35(...TRUNCATED)
| 6
| [-0.007698036264628172,-0.000024437606043647975,-0.20155395567417145,-0.2014216184616089,-0.19169320(...TRUNCATED)
| [-3.576278118089249e-7,0.0,0.0,-1.1920928244535389e-7,-2.3841855067985307e-7,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let's tackle this problem step by step. So, the question is about rotating the graph of y = l(...TRUNCATED)
| "Okay, let me try to figure out this problem. So, the question is about rotating the graph of y = lo(...TRUNCATED)
| 7
| [-0.08929406851530075,-0.000036477376852417365,-0.7578859925270081,-1.6004422903060913,-0.0051341354(...TRUNCATED)
| [-3.576278118089249e-7,-3.576278118089249e-7,0.0,-2.3841855067985307e-7,-4.768370445162873e-7,0.0,0.(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, so I need to figure out who ends up with the largest final answer among Jose, Thuy, and Karee(...TRUNCATED)
| "Okay, let's see. The problem is about Jose, Thuy, and Kareem each starting with the number 10 and p(...TRUNCATED)
| 8
| [-0.030001254752278328,-0.00003099393507000059,-0.17984434962272644,-0.2254335582256317,-0.566813111(...TRUNCATED)
| [-3.576278118089249e-7,0.0,0.0,-2.3841855067985307e-7,-3.576278118089249e-7,0.0,0.0,-2.3841855067985(...TRUNCATED)
|
"<|im_start|>system\nYour role as an assistant involves reconstructing the internal reasoning proces(...TRUNCATED)
| "Okay, let me try to figure out this problem. So, we have a set of consecutive positive integers sta(...TRUNCATED)
| "Okay, let me try to figure out this problem. So, we have a set of consecutive positive integers sta(...TRUNCATED)
| 9
| [-0.07026564329862595,-0.000015258672647178173,-0.3486752510070801,-1.8048397302627563,-0.0022368428(...TRUNCATED)
| [-1.4305104514278355e-6,-0.000030278701160568744,0.0,-2.3841855067985307e-7,-8.940656698541716e-6,0.(...TRUNCATED)
|
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