Question
A student simplified this expression: (33⋅ 2−4)436
(
3
3
⋅
2
−
4
)
4
3
6
using these steps:
Step 1: 312 ⋅ 2−1636
3
12
⋅
2
−
16
3
6
Step 2: 36216
3
6
2
16
(
3
3
⋅
2
−
4
)
4
3
6
using these steps:
Step 1: 312 ⋅ 2−1636
3
12
⋅
2
−
16
3
6
Step 2: 36216
3
6
2
16
Answers
GPT-4o mini
Let's analyze the expression and the steps taken in simplification:
The original expression is \((33 \cdot 2^{-4})^{436}\).
**Step 1: Simplifying the base**
We start with:
\[
33 \cdot 2^{-4}
\]
This can be expressed as:
\[
3^3 \cdot 2^{-4}
\]
So the expression becomes:
\[
(3^3 \cdot 2^{-4})^{436}
\]
Using properties of exponents, we distribute the exponent:
\[
(3^3)^{436} \cdot (2^{-4})^{436} = 3^{3 \cdot 436} \cdot 2^{-4 \cdot 436}
\]
Calculating the powers:
\[
3^{1308} \cdot 2^{-1744}
\]
**Step 2: Expressing the results in simplified terms**
The next step would create a similar structure. We can combine the powers of two or write the expression in a more manageable format:
\[
\frac{3^{1308}}{2^{1744}}
\]
Now let's look at the student's steps:
1. **Step 1**: The student seems to express it as \(3^{12} \cdot 2^{-16}\), which appears incorrect. The calculation \(3^{1308} \cdot 2^{-1744}\) is what we should retain.
2. **Step 2**: It seems they made a mistake again in this simplification.
Therefore, the final simplification should correctly arrive at:
\[
\frac{3^{1308}}{2^{1744}}
\]
If you need further steps or breakdowns, feel free to ask!
The original expression is \((33 \cdot 2^{-4})^{436}\).
**Step 1: Simplifying the base**
We start with:
\[
33 \cdot 2^{-4}
\]
This can be expressed as:
\[
3^3 \cdot 2^{-4}
\]
So the expression becomes:
\[
(3^3 \cdot 2^{-4})^{436}
\]
Using properties of exponents, we distribute the exponent:
\[
(3^3)^{436} \cdot (2^{-4})^{436} = 3^{3 \cdot 436} \cdot 2^{-4 \cdot 436}
\]
Calculating the powers:
\[
3^{1308} \cdot 2^{-1744}
\]
**Step 2: Expressing the results in simplified terms**
The next step would create a similar structure. We can combine the powers of two or write the expression in a more manageable format:
\[
\frac{3^{1308}}{2^{1744}}
\]
Now let's look at the student's steps:
1. **Step 1**: The student seems to express it as \(3^{12} \cdot 2^{-16}\), which appears incorrect. The calculation \(3^{1308} \cdot 2^{-1744}\) is what we should retain.
2. **Step 2**: It seems they made a mistake again in this simplification.
Therefore, the final simplification should correctly arrive at:
\[
\frac{3^{1308}}{2^{1744}}
\]
If you need further steps or breakdowns, feel free to ask!