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

Question

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=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=−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

Answer 1

To determine which expressions are equivalent using the Associative Property, we need to analyze each expression provided. The Associative Property states that the way numbers are grouped in addition or multiplication does not change the result.

Let's examine each response:

1. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y):

   • This is a correct application of the Associative Property as it shows how to regroup the terms in multiplication. So, this expression is equivalent.

2. −3⋅(4x⋅−2)⋅−6y = 18−8x:

   • This requires actual calculation. On the left side: \[-3 \cdot (4x \cdot -2) \cdot -6y = -3 \cdot (-8x) \cdot -6y = 144xy.\]
   • On the right side, \(18 - 8x\) does not equal \(144xy\). So, this expression is not equivalent.

3. −3⋅(4x⋅−2)⋅−6y = −7xy:

   • Again, calculating the left side gives us \(144xy\).
   • The right side does not equal \(144xy\). Therefore, this expression is not equivalent as well.

4. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)⋅−2−6y:

   • This also needs to be worked out. However, without needing calculations, we can conclude that this is structurally incorrect because the right side does not correctly follow the associative property in grouping.

So, the only expression that is clearly equivalent through the Associative Property is the first one:

−3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y).