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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 open paren 4 x minus 2 close paren times negative 6 y is equal to open paren negative 3 times 4 x close paren times negative 2 minus 6 y

−3⋅(4x−2)⋅−6y=−7xy
negative 3 times open paren 4 x minus 2 close paren times negative 6 y is equal to negative 7 x y

−3⋅(4x−2)⋅−6y=(−3⋅4x)(−2⋅−6y)
negative 3 times open paren 4 x minus 2 close paren times negative 6 y is equal to open paren negative 3 times 4 x close paren times open paren negative 2 times negative 6 y close paren

−3⋅(4x−2)⋅−6y=18−8x

1 answer

To determine which expressions are equivalent using the Associative Property, we first need to simplify the left side of each given expression:

Left Side: \[ -3 \cdot (4x - 2) \cdot -6y \]

Using the Associative Property, we can regroup and simplify: \[ = (-3 \cdot -6y) \cdot (4x - 2) \] \[ = 18y \cdot (4x - 2) \] \[ = 18y \cdot 4x - 18y \cdot 2 \] \[ = 72xy - 36y \]

Now let's compare this with the options:

  1. Option 1: \[ (−3 \cdot 4x) \cdot -2 - 6y \] Simplifying this: \[ = -12x \cdot -2 - 6y = 24x - 6y \quad (Not equivalent) \]

  2. Option 2: \[ -7xy \] (Not equivalent to \(72xy - 36y\))

  3. Option 3: \[ (−3 \cdot 4x)(−2 \cdot -6y) \] Simplifying this: \[ = -12x \cdot 12y = 144xy \quad (Not equivalent to \(72xy - 36y\)) \]

  4. Option 4: \[ 18 - 8x \] (Not equivalent to \(72xy - 36y\))

After evaluating all options, none of the given expressions are equivalent to the simplified version of the left side \(72xy - 36y\).

In conclusion, none of the responses provided are equivalent to \(-3 \cdot (4x - 2) \cdot -6y\).