Question
Use the Associative Property to determine which expressions are equivalent.(1 point)
Responses
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals left parenthesis negative 3 times 4 x right parenthesis left parenthesis negative 2 times negative 6 y right parenthesis
−3⋅(4x⋅−2)⋅−6y=−7xy
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals negative 7 x y
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals left parenthesis negative 3 times 4 x right parenthesis times negative 2 minus 6 y
−3⋅(4x⋅−2)⋅−6y=18−8x
Responses
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals left parenthesis negative 3 times 4 x right parenthesis left parenthesis negative 2 times negative 6 y right parenthesis
−3⋅(4x⋅−2)⋅−6y=−7xy
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals negative 7 x y
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals left parenthesis negative 3 times 4 x right parenthesis times negative 2 minus 6 y
−3⋅(4x⋅−2)⋅−6y=18−8x
Answers
GPT-4o mini
To determine which expressions are equivalent using the Associative Property, let’s evaluate the first response and then check the others.
1. **Response 1**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)
This uses the Associative Property correctly. We can rewrite the left side by grouping \( (4x \cdot -2) \) first and multiplying everything out. The right side is grouping the factors differently but ends up representing the same multiplication. So, this is equivalent.
2. **Response 2**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\)
This states that the left side is equivalent to \(-7xy\). We’ll need to calculate the left side:
- First, evaluate \( 4x \cdot -2 = -8x \).
- Now, calculate \(-3 \cdot -8x \cdot -6y = -3 \cdot -8 \cdot -6 \cdot x \cdot y = -144xy\).
Clearly, this does not equal \(-7xy\). Therefore, they are not equivalent.
3. **Response 3**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 - 6y\)
The left side evaluates to, as above, \(-144xy\). The right side doesn't look like a direct application of Associative Property as it contains a subtraction involving \(-6y\). Therefore, we can evaluate:
- The expression \((-3 \cdot 4x) \cdot -2\) will yield \(24x\).
- Then adding \(-6y\) does not lead to \(-144xy\), so these are not equivalent.
4. **Response 4**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)
Again, since the left side evaluates to \(-144xy\) and the right side \(18 - 8x\) is a polynomial in different variables\( (x \text{ and } y)\), these are not equivalent.
### Conclusion:
**Only Response 1 is equivalent** using the Associative Property.
1. **Response 1**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)
This uses the Associative Property correctly. We can rewrite the left side by grouping \( (4x \cdot -2) \) first and multiplying everything out. The right side is grouping the factors differently but ends up representing the same multiplication. So, this is equivalent.
2. **Response 2**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\)
This states that the left side is equivalent to \(-7xy\). We’ll need to calculate the left side:
- First, evaluate \( 4x \cdot -2 = -8x \).
- Now, calculate \(-3 \cdot -8x \cdot -6y = -3 \cdot -8 \cdot -6 \cdot x \cdot y = -144xy\).
Clearly, this does not equal \(-7xy\). Therefore, they are not equivalent.
3. **Response 3**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 - 6y\)
The left side evaluates to, as above, \(-144xy\). The right side doesn't look like a direct application of Associative Property as it contains a subtraction involving \(-6y\). Therefore, we can evaluate:
- The expression \((-3 \cdot 4x) \cdot -2\) will yield \(24x\).
- Then adding \(-6y\) does not lead to \(-144xy\), so these are not equivalent.
4. **Response 4**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)
Again, since the left side evaluates to \(-144xy\) and the right side \(18 - 8x\) is a polynomial in different variables\( (x \text{ and } y)\), these are not equivalent.
### Conclusion:
**Only Response 1 is equivalent** using the Associative Property.