Question
A student simplified this expression: (3^3⋅ 2^−4)^4/3^6 using these steps:Step 1: 312 ⋅ 2−1636 Step 2: 36216 Step 3: Answer is 0.011(1 point)ResponsesThere 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)436Step 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 powerStep 2 had the mistake. It should have been 32 =9Step 2 had the mistake. It should have been 32 =9Step 3 had the mistake. It should have been 3−2 = 1−2 = 1
Answers
GPT-4o mini
To simplify the expression \((3^3 \cdot 2^{-4})^{4/3} \) divided by \(3^6\), let's start from the beginning and check the steps one by one.
The expression is:
\[
\frac{(3^3 \cdot 2^{-4})^{4/3}}{3^6}
\]
### Step 1: Simplify the numerator
Apply the exponent to both parts inside the parentheses:
\[
(3^3)^{4/3} \cdot (2^{-4})^{4/3} = 3^{(3 \cdot \frac{4}{3})} \cdot 2^{(-4 \cdot \frac{4}{3})}
\]
This simplifies to:
\[
3^4 \cdot 2^{-\frac{16}{3}}
\]
### Step 2: Write the full expression
Now, we substitute this back into the original expression:
\[
\frac{3^4 \cdot 2^{-\frac{16}{3}}}{3^6}
\]
### Step 3: Simplify the powers of 3
Using the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
3^{4-6} \cdot 2^{-\frac{16}{3}} = 3^{-2} \cdot 2^{-\frac{16}{3}} = \frac{1}{3^2} \cdot \frac{1}{2^{\frac{16}{3}}} = \frac{1}{9 \cdot 2^{\frac{16}{3}}}
\]
### Step 4: Finding a decimal approximation (if needed)
The expression can be further simplified to obtain a decimal approximation, but the main evaluation here shows there is a mistake in the original student's simplification.
Let's analyze the provided choices against our derived result:
1. **There is no mistake. The answer is correct.** - **Incorrect** (we found mistakes).
2. **Step 1 had the mistake. It should have been (6−4)436** - **Correct** (the exponent should have been \( (3-6)\) not the simplified one they provided).
3. **Step 1 had the mistake. It should have been the fraction with the numerator open paren 6 to the negative 4 power close to the 4th power and denominator 3 to the 6th power.** - **Incorrect** (while a valid simplification, this is not the route taken).
4. **Step 2 had the mistake. It should have been \(3^2 = 9\)** - Partially true but not the main mistake.
5. **Step 3 had the mistake. It should have been \(3^{-2} = 1^{-2} = 1\)** - Incorrect, as \(3^{-2}\) is definitely not equal to \(1\).
Thus, the most accurate response is:
**Step 1 had the mistake. It should have been \((3^{4/3})\cdot(2^{-\frac{16}{3}})\) and then simplifying correctly.**
The expression is:
\[
\frac{(3^3 \cdot 2^{-4})^{4/3}}{3^6}
\]
### Step 1: Simplify the numerator
Apply the exponent to both parts inside the parentheses:
\[
(3^3)^{4/3} \cdot (2^{-4})^{4/3} = 3^{(3 \cdot \frac{4}{3})} \cdot 2^{(-4 \cdot \frac{4}{3})}
\]
This simplifies to:
\[
3^4 \cdot 2^{-\frac{16}{3}}
\]
### Step 2: Write the full expression
Now, we substitute this back into the original expression:
\[
\frac{3^4 \cdot 2^{-\frac{16}{3}}}{3^6}
\]
### Step 3: Simplify the powers of 3
Using the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
3^{4-6} \cdot 2^{-\frac{16}{3}} = 3^{-2} \cdot 2^{-\frac{16}{3}} = \frac{1}{3^2} \cdot \frac{1}{2^{\frac{16}{3}}} = \frac{1}{9 \cdot 2^{\frac{16}{3}}}
\]
### Step 4: Finding a decimal approximation (if needed)
The expression can be further simplified to obtain a decimal approximation, but the main evaluation here shows there is a mistake in the original student's simplification.
Let's analyze the provided choices against our derived result:
1. **There is no mistake. The answer is correct.** - **Incorrect** (we found mistakes).
2. **Step 1 had the mistake. It should have been (6−4)436** - **Correct** (the exponent should have been \( (3-6)\) not the simplified one they provided).
3. **Step 1 had the mistake. It should have been the fraction with the numerator open paren 6 to the negative 4 power close to the 4th power and denominator 3 to the 6th power.** - **Incorrect** (while a valid simplification, this is not the route taken).
4. **Step 2 had the mistake. It should have been \(3^2 = 9\)** - Partially true but not the main mistake.
5. **Step 3 had the mistake. It should have been \(3^{-2} = 1^{-2} = 1\)** - Incorrect, as \(3^{-2}\) is definitely not equal to \(1\).
Thus, the most accurate response is:
**Step 1 had the mistake. It should have been \((3^{4/3})\cdot(2^{-\frac{16}{3}})\) and then simplifying correctly.**