To determine which expressions are equivalent using the Associative Property, we must first understand what the property means. The Associative Property states that the way numbers are grouped in an expression does not change their product or sum. In multiplication, for example, we can say:
\((a \times b) \times c = a \times (b \times c)\)
Now, let's analyze each expression:
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Expression 1: \(-3 \times (4x \times -2) \times -6y\)
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Expression 2: \((-3 \times 4x) \times -2 - 6y\)
- This appears to be a different structure entirely because of the subtraction of \(6y\). Hence, this expression does not simplify to the same form as Expression 1.
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Expression 3: \((-3 \times 4x)(-2 \times -6y)\)
- This uses a different grouping and combines three factors, which does not directly compare to Expression 1 through associative grouping. Therefore, we need to compute this separately.
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Expression 4: \(-7xy\)
- This is a completely different polynomial expression, which does not relate to the structure given in Expression 1.
Evaluation:
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Expression 1:
- Calculate individual parts: \(-3 \times (4x \times -2) \times -6y = -3 \times (-8x) \times -6y = 24xy\)
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Expression 2:
- This simplifies to \((-12x) - 6y\), which does not relate to expression 1.
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Expression 3:
- Calculate: \[ (-3 \times 4x)(-2 \times -6y) = (-12x)(12y) = -144xy \]
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Expression 4:
- The expression \(-7xy\) does not correspond to the computed values from Expression 1 or Expression 3.
Result:
Based on this analysis:
- Expression 1 \((-3 \times (4x \times -2) \times -6y)\) corresponds to \(24xy\).
- None of the other expressions simplify to this result.
Thus, none of the expressions you provided are equivalent to each other.