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

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=−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=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

1 answer

To evaluate which expressions are equivalent using the Associative Property, we'll analyze each response.

  1. First expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \] This uses the Associative Property correctly, as it groups the multiplication of terms, but it does not establish if both sides are equal numerically.

  2. Second expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \] This implies that the left side evaluates to -7xy. However, we first need to simplify the left side to confirm or deny this equality.

  3. Third expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \] Similar to the second expression, we need to simplify the left side to determine whether this statement is true.

  4. Fourth expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \] This rearrangement maintains the same terms and shows another application of the Associative Property.

Let's simplify \( -3 \cdot (4x \cdot -2) \cdot -6y \):

  1. Calculate \( 4x \cdot -2 = -8x \).
  2. Now substitute: \[ -3 \cdot -8x \cdot -6y = 24x \cdot -6y = -144xy \]

Now we evaluate each of the expressions:

  • The first one doesn't numerically verify equality; it's just a rearrangement.
  • The second states \( -144xy = -7xy \) which is false.
  • The third states \( -144xy = 18 - 8x \), which is also false.
  • The fourth does not establish a numerical equality but proves an associative rearrangement.

Thus, the only response that holds true in terms of equality (considering associative property application and product evaluation) is the first expression correctly states an application of the Associative Property, but none confirm numerical equivalency since they differ in their resultant products.

Therefore, there are no equivalent expressions among the provided options based strictly on the equality derived from simplification.