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Bot GPT 4-o mini To determine which expressions are equivalent using the Associative Property, we'll first focus on the structure of the expression in A:
The Associative Property states that the way numbers are grouped in multiplication does not change their product. This means that for any numbers \( a, b, \) and \( c \):
\[
(a \cdot b) \cdot c = a \cdot (b \cdot c)
\]
Let's analyze each response:
**A.**
\[
-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot (-2 \cdot -6y)
\]
This uses the Associative Property correctly by changing the grouping of the terms in the product. This statement is equivalent.
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**B.**
\(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)
To evaluate this, we will calculate \( -3 \cdot (4x \cdot -2) \cdot -6y \) and compare it to \( 18 - 8x \). Calculating it directly, we get:
\[
-3 \cdot (4x \cdot -2) \cdot -6y = -3 \cdot (-8x) \cdot -6y = 24xy
\]
Clearly, \( 24xy \neq 18 - 8x\), so B is not equivalent.
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**C.**
\(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\)
Using the same calculation from above, we found \( -3 \cdot (4x \cdot -2) \cdot -6y = 24xy\), and \( 24xy \neq -7xy\), so C is not equivalent.
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**D.**
\(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 - 6y\)
This expression does not utilize the Associative Property correctly, as it introduces an addition sign instead of maintaining multiplication. On evaluation, we see that it doesn't simplify to the left side, therefore it's not equivalent.
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Thus, the only expression that is equivalent by the Associative Property is:
**A.** \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot (-2 \cdot -6y)\).