To use the Associative Property and determine which expressions are equivalent, we need to understand the structure of the expressions provided. The Associative Property states that the way in which numbers are grouped in multiplication does not change their product.
Let's evaluate each expression.
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Expression 1: \[ -3 \cdot (4x \cdot -2) \cdot -6y \] We can rearrange it using the associative property as: \[ (-3 \cdot 4x \cdot -2) \cdot -6y \] or \[ -3 \cdot ((4x \cdot -2) \cdot -6y) \] This is equivalent to whatever grouping we do, the product remains the same.
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Expression 2: \[ -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \] To verify if this expression is equivalent, we would evaluate:
- Calculate \(-3 \cdot 4x \cdot -2 = 24x\).
- Now multiply with \(-6y\): \(24x \cdot -6y = -144xy\). This shows that \( -3 \cdot (4x \cdot -2) \cdot -6y \neq -7xy\).
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Expression 3: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \] This one uses the associative property correctly since we are just regrouping without changing the multiplicative outcome.
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Expression 4: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \] We can check if this is equivalent to our first expression; rearranging gives the same outcome as the expression remains equivalent by the same rules of multiplication.
From our analysis, Expression 2 is NOT equivalent. Expressions 1, 3, and 4 are equivalent as they maintain the same outcome when evaluated, utilizing the associative property correctly.
Therefore, the expressions that are equivalent are:
- \(-3 \cdot (4x \cdot -2) \cdot -6y\)
- \((-3 \cdot 4x) \cdot -2 \cdot -6y\)
- \((-3 \cdot 4x)(-2 \cdot -6y)\)
Thus, the expressions equivalent to the first one are expressions 3 and 4.
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