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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=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=−7xy

1 answer

To solve the problem and determine which expressions are equivalent using the Associative Property, we will analyze each provided expression. The Associative Property states that the way numbers are grouped in multiplication does not change their product. In other words, for any real numbers \(a\), \(b\), and \(c\), \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).

  1. Expression A: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)

    • First, we can evaluate \(-3 \cdot (4x \cdot -2)\):

      • This simplifies to \(-3 \cdot -8x = 24x\).

    • Now multiply by \(-6y\):

      • \(24x \cdot -6y = -144xy\).

    • Now evaluate \((-3 \cdot 4x)(-2 \cdot -6y)\):

      • \(-3 \cdot 4x = -12x\) and \(-2 \cdot -6y = 12y\).

    • Now multiply:

      • \(-12x \cdot 12y = -144xy\).

    • Both sides simplify to \(-144xy\), so this expression is equivalent.

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

    • We already calculated the left side: \(-144xy\).
    • The right side, \(18 - 8x\), does not simplify to \(-144xy\).
    • Therefore, this expression is not equivalent.

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

    • Again, we have: \(-144xy\) on the left side.
    • The right side can be analyzed:
      • First, find \(-3 \cdot 4x = -12x\).
      • But \((-12x) \cdot -2 - 6y\) does not yield \(-144xy\).
    • Thus, this expression is not equivalent as well.

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

    • We have \(-144xy\) on the left side.
    • The right side is simply \(-7xy\).
    • These do not match, so this expression is also not equivalent.

In conclusion, the only equivalent expression using the Associative Property is Expression A: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\).