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 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=18−8x

Answer 1

To determine which expressions are equivalent using the Associative Property, we need to understand that the Associative Property states that the way in which numbers are grouped in multiplication or addition does not affect the overall product or sum.

Let's assess the listed expressions to see which can be transformed into each other based on this principle:

1. \( -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \)
This seems to equate to something different, so let's set it aside.

2. \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \)
This is an application of the Associative Property and is equivalent.

3. \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \)
This expression is also equivalent as it reorganizes the grouping.

4. \( -3 \cdot (4x \cdot -2) \cdot -6y = ( -3 \cdot 4x) \cdot -2 - 6y \)
This expression is not equivalent due to the structure of the right side (it looks like it might not stick to multiplication).

5. \( -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \)
This expression is also not equivalent as it equates a multiplication to a sum.

From the analysis above, the following expressions are equivalent:
- \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \)
- \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \)

So, the valid equivalent expressions based on the Associative Property are the second and third responses.