Question

Use the Associative Property to determine which expressions are equivalent.(1 point)
Responses

−3⋅(4x⋅−2)⋅−6y=18−8x
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals 18 minus 8 x

−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=(−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=−7xy
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals negative 7 x y

Answer 1

To determine which expressions are equivalent using the Associative Property, we can analyze the provided responses. The Associative Property states that the way in which numbers are grouped in multiplication or addition does not change their product or sum.

Let's analyze each response in detail:

1. **Response 1**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)
This equation implies an equality that would need further simplification to verify; however, it does not directly demonstrate the Associative Property.

2. **Response 2**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)
This is an application of the Associative Property, as it demonstrates the regrouping of factors. It shows that the same numbers are multiplied in a different order without changing the result.

3. **Response 3**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 - 6y\)
This expression does not apply the Associative Property correctly, as it suggests subtraction in the grouping, which does not maintain the equality verified by the original multiplication.

4. **Response 4**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\)
This implies an equality that would also need verification through simplification, but it again does not showcase the usage of the Associative Property.

From the analysis, **Response 2** correctly demonstrates the Associative Property by showing that multiplication can be regrouped, thus making it equivalent.