Question
A student simplified this expression: (3 to the power of three times 2 to the power of negative four) to the power of four divided by 3 to the power of six using these steps:
Step 1: 3 to the power of twelve times 2 to the power of negative sixteen divided by 3 to the power of six
Step 2: 3 to the power of six divided by 2 to the power of sixteen
Step 3: ANSWER is 0.011
There is no mistake. The answer is correct.
Step 1 had the mistake
Step 2 had the mistake
Step 3 had the mistake
Step 1: 3 to the power of twelve times 2 to the power of negative sixteen divided by 3 to the power of six
Step 2: 3 to the power of six divided by 2 to the power of sixteen
Step 3: ANSWER is 0.011
There is no mistake. The answer is correct.
Step 1 had the mistake
Step 2 had the mistake
Step 3 had the mistake
Answers
GPT-4o mini
Let's analyze the student's work step by step to identify where the mistakes occurred.
1. **Starting Expression:** \((3^{3 \cdot 2^{-4}})^4 \div 3^6\)
We can simplify the expression step by step.
2. **Step 1: Simplification of the exponent**
- The expression inside the parentheses can be rewritten as \(3^{3} \cdot 2^{-4}\).
- Raising this to the 4th power gives us:
\[
3^{3 \cdot 4} \cdot 2^{-4 \cdot 4} = 3^{12} \cdot 2^{-16}.
\]
- Dividing by \(3^{6}\):
\[
\frac{3^{12} \cdot 2^{-16}}{3^{6}} = 3^{12-6} \cdot 2^{-16} = 3^{6} \cdot 2^{-16}.
\]
- **So Step 1 was simplified correctly.**
3. **Step 2: Continue simplifying**
- The expression \(3^{6} \cdot 2^{-16}\) does not simplify directly to \(3^{6} \div 2^{16}\). It actually remains \(3^{6} \cdot 2^{-16}\).
- Therefore, the calculation in Step 2 has a mistake:
- The correct form, before arriving at the numerical calculation, is:
\[
3^{6} \cdot \frac{1}{2^{16}}.
\]
4. **Step 3: Numerical calculation**
- The final numerical evaluation should be:
\[
\frac{3^{6}}{2^{16}}.
\]
- Calculating this explicitly:
- \(3^{6} = 729\),
- \(2^{16} = 65536\),
- So, the final evaluation gives:
\[
\frac{729}{65536} \approx 0.011.
\]
In summary, the mistakes are as follows:
- **Step 1:** Was performed correctly.
- **Step 2:** Had the mistake; it should have kept the multiplication form rather than the division.
- **Step 3:** Is incorrect only in its derivation chain but arrives at the correct numerical answer.
Thus, the conclusions about the mistakes:
- **Step 2 had the mistake.**
- **Step 3 has a correct numerical output (0.011).**
1. **Starting Expression:** \((3^{3 \cdot 2^{-4}})^4 \div 3^6\)
We can simplify the expression step by step.
2. **Step 1: Simplification of the exponent**
- The expression inside the parentheses can be rewritten as \(3^{3} \cdot 2^{-4}\).
- Raising this to the 4th power gives us:
\[
3^{3 \cdot 4} \cdot 2^{-4 \cdot 4} = 3^{12} \cdot 2^{-16}.
\]
- Dividing by \(3^{6}\):
\[
\frac{3^{12} \cdot 2^{-16}}{3^{6}} = 3^{12-6} \cdot 2^{-16} = 3^{6} \cdot 2^{-16}.
\]
- **So Step 1 was simplified correctly.**
3. **Step 2: Continue simplifying**
- The expression \(3^{6} \cdot 2^{-16}\) does not simplify directly to \(3^{6} \div 2^{16}\). It actually remains \(3^{6} \cdot 2^{-16}\).
- Therefore, the calculation in Step 2 has a mistake:
- The correct form, before arriving at the numerical calculation, is:
\[
3^{6} \cdot \frac{1}{2^{16}}.
\]
4. **Step 3: Numerical calculation**
- The final numerical evaluation should be:
\[
\frac{3^{6}}{2^{16}}.
\]
- Calculating this explicitly:
- \(3^{6} = 729\),
- \(2^{16} = 65536\),
- So, the final evaluation gives:
\[
\frac{729}{65536} \approx 0.011.
\]
In summary, the mistakes are as follows:
- **Step 1:** Was performed correctly.
- **Step 2:** Had the mistake; it should have kept the multiplication form rather than the division.
- **Step 3:** Is incorrect only in its derivation chain but arrives at the correct numerical answer.
Thus, the conclusions about the mistakes:
- **Step 2 had the mistake.**
- **Step 3 has a correct numerical output (0.011).**