Question

To determine which expressions are equivalent by using the Associative Property, we need to understand what the Associative Property states. The Associative Property allows us to group numbers in different ways when performing addition or multiplication. Specifically, for multiplication, it states that \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).

Now let's analyze the expressions given:

1. **−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)**
- This expression relies more on the Distributive Property and does not seem to correctly apply the Associative Property. They are not equivalent because the grouping and order of multiplication differ.

2. **−3⋅(4x⋅−2)⋅−6y=18−8x**
- This seems to imply that the left-hand expression simplifies to a numerical value involving \( x \). This is likely incorrect without further calculations or specific rates for the variables. Thus, not equivalent.

3. **−3⋅(4x⋅−2)⋅−6y=−7xy**
- Like the previous expression, this suggests a specific equality which likely is not true unless specific values for \( x \) and \( y \) are provided. No indication this holds.

4. **−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y**
- This expression attempts to manipulate the grouping of the original multiplication and does use the Associative Property correctly. This is likely the one that maintains equivalence.

So, the only expression that uses the Associative Property correctly and remains equivalent to the left-hand side is:

**−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y**.

This is the correct response based on the Associative Property.