Question
A student simplified this expression: (3 superscript3 ⋅ 2 − superscript4) superscript4/3 superscript6
using these steps:
Step 1: 312 ⋅ 2−1636
Step 2: 36216
Step 3: Answer is 0.011
(1 point)
Responses
There is no mistake. The answer is correct.
There is no mistake. The answer is correct.
Step 1 had the mistake. It should have been (6−4)436
Step 1 had the mistake. It should have been the fraction with numerator open paren 6 to the negative 4 power close paren to the 4th power and denominator 3 to the 6th power
Step 2 had the mistake. It should have been 32 =9
Step 2 had the mistake. It should have been 32 =9
Step 3 had the mistake. It should have been 3−2 = 1−2 = 1
using these steps:
Step 1: 312 ⋅ 2−1636
Step 2: 36216
Step 3: Answer is 0.011
(1 point)
Responses
There is no mistake. The answer is correct.
There is no mistake. The answer is correct.
Step 1 had the mistake. It should have been (6−4)436
Step 1 had the mistake. It should have been the fraction with numerator open paren 6 to the negative 4 power close paren to the 4th power and denominator 3 to the 6th power
Step 2 had the mistake. It should have been 32 =9
Step 2 had the mistake. It should have been 32 =9
Step 3 had the mistake. It should have been 3−2 = 1−2 = 1
Answers
GPT-4o mini
Let's analyze the simplification process step by step based on the original expression:
\[
\left(3^3 \cdot 2^{-4}\right)^{\frac{4}{3}} \cdot 3^6
\]
### Step 1 Analysis
The expression simplifies as follows:
1. Calculate \(3^3\) and \(2^{-4}\):
\[
3^3 = 27, \quad 2^{-4} = \frac{1}{16}
\]
So,
\[
3^3 \cdot 2^{-4} = 27 \cdot \frac{1}{16} = \frac{27}{16}
\]
2. Raise this result to the power of \(\frac{4}{3}\):
\[
\left(\frac{27}{16}\right)^{\frac{4}{3}} = \frac{27^{\frac{4}{3}}}{16^{\frac{4}{3}}}
\]
Here, \(27 = 3^3 \Rightarrow 27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^4 = 81\).
And \(16 = 2^4 \Rightarrow 16^{\frac{4}{3}} = (2^4)^{\frac{4}{3}} = 2^{\frac{16}{3}}\).
3. The expression can then look like:
\[
\frac{81}{2^{\frac{16}{3}}}
\]
4. Now, also include \(3^6\):
\[
\frac{81 \cdot 3^6}{2^{\frac{16}{3}}}
\]
At this point, the expression is not yet simplified into the form in the student's steps, which indicates Step 1 had an initial mistake.
### Conclusion of Step 1
The correct Step 1 should indeed have been about simplifying combining \(3^3 \cdot 2^{-4}\) correctly, and avoid the intermediate format that led to a loss in powers. Therefore, **Step 1 had the mistake. It should have been the fraction with numerator \((3^3 \cdot 2^{-4})^{4/3}\) to reflect correctly.**
### Selecting Responses
- The correct response is: **Step 1 had the mistake. It should have been the fraction with numerator \((3^3 \cdot 2^{-4})^{4/3}\) and denominator \(3^6\)**.
### Other Mistakes
Step 2 and Step 3 would likely contain further inaccuracies based on the misunderstanding in Step 1. Without fully exploring those, the above answer remains the key focus for providing clarity and guidance regarding the simplification.
\[
\left(3^3 \cdot 2^{-4}\right)^{\frac{4}{3}} \cdot 3^6
\]
### Step 1 Analysis
The expression simplifies as follows:
1. Calculate \(3^3\) and \(2^{-4}\):
\[
3^3 = 27, \quad 2^{-4} = \frac{1}{16}
\]
So,
\[
3^3 \cdot 2^{-4} = 27 \cdot \frac{1}{16} = \frac{27}{16}
\]
2. Raise this result to the power of \(\frac{4}{3}\):
\[
\left(\frac{27}{16}\right)^{\frac{4}{3}} = \frac{27^{\frac{4}{3}}}{16^{\frac{4}{3}}}
\]
Here, \(27 = 3^3 \Rightarrow 27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^4 = 81\).
And \(16 = 2^4 \Rightarrow 16^{\frac{4}{3}} = (2^4)^{\frac{4}{3}} = 2^{\frac{16}{3}}\).
3. The expression can then look like:
\[
\frac{81}{2^{\frac{16}{3}}}
\]
4. Now, also include \(3^6\):
\[
\frac{81 \cdot 3^6}{2^{\frac{16}{3}}}
\]
At this point, the expression is not yet simplified into the form in the student's steps, which indicates Step 1 had an initial mistake.
### Conclusion of Step 1
The correct Step 1 should indeed have been about simplifying combining \(3^3 \cdot 2^{-4}\) correctly, and avoid the intermediate format that led to a loss in powers. Therefore, **Step 1 had the mistake. It should have been the fraction with numerator \((3^3 \cdot 2^{-4})^{4/3}\) to reflect correctly.**
### Selecting Responses
- The correct response is: **Step 1 had the mistake. It should have been the fraction with numerator \((3^3 \cdot 2^{-4})^{4/3}\) and denominator \(3^6\)**.
### Other Mistakes
Step 2 and Step 3 would likely contain further inaccuracies based on the misunderstanding in Step 1. Without fully exploring those, the above answer remains the key focus for providing clarity and guidance regarding the simplification.