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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 times negative 2 minus 6 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=−7xy

1 answer

To determine which expressions are equivalent using the Associative Property, we need to examine the expressions provided.

The Associative Property states that for addition or multiplication, the way numbers are grouped does not change their value. This means that:

  • For multiplication: \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \)
  • For addition: \( a + (b + c) = (a + b) + c \)

Now, let's analyze the responses one by one:

  1. Response 1: \(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)

    • This equation compares a product with a linear expression (18 - 8x), and they are not equal. So, this is not valid.

  2. Response 2: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y\)

    • This uses the associative property correctly (rearranging the grouping of terms).

  3. Response 3: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)

    • This uses the associative property too, but the right side involves grouping the first part and a separate multiplication for -2 and -6y.

  4. Response 4: \(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\)

    • This compares a product with another expression (-7xy), and they are not equal.

Given that, the responses that correctly use the Associative Property and maintain equivalence of the expressions are Response 2 and Response 3. Therefore, they're equivalent in terms of using the associative property correctly.

Conclusion: The expressions in Response 2 and Response 3 are equivalent based on the Associative Property.