Let's analyze the student's steps to find the mistake.
The original expression is \( 33 \cdot 3^{-2} \cdot (36)^2 \).
Let's simplify it step by step:
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Begin with the original expression: \[ 33 \cdot 3^{-2} \cdot (36)^2 \]
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Recognize \( 36 = 6^2 \), so: \[ (36)^2 = 36^2 = (6^2)^2 = 6^4 \]
Now we note that \( 33 = 3^3 \): \[ 3^3 \cdot 3^{-2} \cdot 6^4 \]
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Combine the powers of 3: \[ 3^{3 - 2} = 3^1 = 3 \]
Thus, we now have: \[ 3 \cdot 6^4 \]
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Using the value of \( 6 \) and simplifying we get: \[ 6^4 = (2 \cdot 3)^4 = 2^4 \cdot 3^4 = 16 \cdot 81 = 1296 \]
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Now substitute back to find: \[ ( 3 \cdot 1296 = 3888 \]
Now, let’s go through the steps provided by the student:
Step 1: Claim was: \[ 3^{-6} \cdot 3^{12} \] This seems to be simplifying incorrectly, since the initial terms should simplify to \( 3^{3-2} \), not this combination.
Step 2: Claims: \[ 1^3 \cdot 3^{12} \] This is also incorrect as it doesn't follow logically from the first step.
Step 3: Claims: \[ 1^3/8 \] Again not correct as it doesn't follow the required calculations.
Based on these observations:
- The first step (Step 1) contains a mistake. The correct simplification for Step 1 should have been \( 3^{3-2} \cdot 6^4 \) instead.
So the response would be: Step 1 had the mistake. Correct answer to this step is \( 3^{1} \cdot 6^4 \).