The Associative Property states that the way numbers are grouped in multiplication does not change the product. In other words, for any numbers \( a, b, c \):
\[ a \times (b \times c) = (a \times b) \times c \]
Let's evaluate each of the provided expressions using the Associative Property.
Expression A:
\[ -3*(4x*-2)*-6y=18-8x \] This expression equates two different types of expressions (one being a product and the other a difference), so we will not consider this expression as equivalent based on the Associative Property since they are not even both products.
Expression B:
\[ -3*(4x*-2)*-6y=-7xy \] Similar to A, this is again two different types of expressions (one a product and the other a single term). So it won't be equivalent through the Associative Property.
Expression C:
\[ -3*(4x*-2)-6y=(-34x)-2-6y \] Here, the left side can be regrouped using the Associative Property, but then it needs further simplification. To assess equivalence, let's evaluate the product on the left side:
- Calculate \( 4x \times -2 = -8x \).
- Now, the left becomes \( -3 \times (-8x) \times -6y = 24xy \).
- The right can also be simplified: \( (-3 \times 4x) = -12x \), and \( -2 \times -6y = 12y \).
- Now evaluate this right side: \((-12x)(12y) = -144xy\).
Since \( 24xy \neq -144xy \), this expression does not hold equivalence.
Expression D:
\[ -3*(4x*-2)-6y=(-34x)(-2*-6y) \] Let's evaluate both sides:
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The left side simplifies as before: \[ = -3 \times (-8x) \times -6y = 24xy \]
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The right side:
- Calculate \(-3 \times 4x = -12x\).
- Next: \(-2 \times -6y = 12y\).
- Evaluate \( (-12x)(12y) = -144xy \).
Again, since \( 24xy \neq -144xy \), this expression does not hold equivalence.
In conclusion, based on the Associative Property of multiplication, none of the expressions A, B, C, or D are equivalent.