To determine which expressions are equivalent using the Associative Property, we will analyze the first expression:
- Original Expression: \(-3 \cdot (4x \cdot -2) \cdot -6y\)
The Associative Property states that the way numbers are grouped in multiplication does not change the product. So, we can rearrange the grouping:
- Using Associative Property: \[-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot (-2 \cdot -6y)\]
This matches exactly with the first expression in the list of responses, which states: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)
Thus, the first expression is equivalent based on the Associative Property.
Now, let's look at the other provided expressions:
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Second Expression: \(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\) — This is likely not equivalent as it simplifies the left side in a specific way.
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Third Expression: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y\) — Here, the grouping is not consistent, and it's not in line with the Associative Property.
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Fourth Expression: \(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\) — This also likely simplifies incorrectly.
Conclusion: The only equivalent expression based on the Associative Property is: −3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y).