Question

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

−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
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)

Answer 1

To determine which of the given expressions are equivalent using the Associative Property, let's analyze each expression step by step.

1. **Expression 1**:
\(-3 \cdot (4x \cdot -2) \cdot -6y\)
This can be simplified by first calculating \(4x \cdot -2\) which is \(-8x\). So we have:
\(-3 \cdot -8x \cdot -6y = -3 \cdot 48xy = -144xy\).

2. **Expression 2**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y\)
This expression is aimed at regrouping terms. Calculating each part gives:
\((-12x) \cdot -2 \cdot -6y\). Multiplying these:
\(-12x \cdot -2 = 24x\), then \(24x \cdot -6y = -144xy\).

3. **Expression 3**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)
From our earlier calculations, we determined that the left side simplifies to \(-144xy\). The expression \(18 - 8x\) does not equal \(-144xy\), so these are not equivalent.

4. **Expression 4**:
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)
This expression also reorganizes the multiplication. The left side simplifies to \(-144xy\) and calculating \((-3 \cdot 4x) \cdot (-2 \cdot -6y)\) gives:
\((-12x) \cdot 12y = 144xy\) (it appears there's a mistake there). The left side yields \(-144xy\), and here we get \(144xy\) which means they do not match either.

Thus, the expressions that are equivalent based on our calculations are:

**The first and second expressions are equivalent:**

- \(-3 \cdot (4x \cdot -2) \cdot -6y\)
- \((-3 \cdot 4x) \cdot -2 \cdot -6y\)

In conclusion, the equivalent expression based on the associative property is the first and second expressions only.