Let's revisit the original expression and the student's steps to identify any mistakes.
The original expression is:
\[ (33 \cdot 2^{-4})^{436} \]
Step 1:
The student wrote
\[ 312 \cdot 2^{-1636} \]
This indicates they likely miscalculated the power or the distribution of terms; with the power of the expression \( (a \cdot b)^n = a^n \cdot b^n \). Correctly applying this rule, we have:
\[ (3^3)^{436} \cdot (2^{-4})^{436} = 3^{3 \cdot 436} \cdot 2^{-4 \cdot 436} = 3^{1308} \cdot 2^{-1744} \]
So, Step 1 had the mistake.
The correct simplification should be:
\[ 3^{1308} \cdot 2^{-1744} \]
Step 2:
Then, the student simplified this further to \( 36216 \). Given the previous simplification was incorrect, this step has a mistake too.
The expression \( 3^{1308} \cdot 2^{-1744} \) does not simplify to \( 36216 \).
Step 3:
The student concludes with \( 0.011 \), which is also incorrect based on the previous simplification steps.
Upon reviewing, we can conclude that:
- Step 1 was incorrect.
- The answer was incorrectly derived from the previous steps.
The correct assertion is:
Step 1 had the mistake. It should have been \( (3^3 \cdot 2^{-4})^{436} \) which simplifies to \( 3^{1308} \cdot 2^{-1744} \).
So, the appropriate choice is:
Step 1 had the mistake. It should have been \( (3^3 \cdot 2^{-4})^{436} \).