Use the Associative Property to determine which expressions are equivalent.

Question

−3⋅(4x⋅−2)⋅−6y=−7xy

−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)

−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y

−3⋅(4x⋅−2)⋅−6y=18−8x

Answer 1

To determine which expressions are equivalent using the Associative Property of multiplication, we can analyze the given expressions.

The Associative Property states that when multiplying numbers, the way in which the numbers are grouped does not affect the product. In other words, \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).

Let's break down the expressions given:

1. Given Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y \]

2. First Comparison: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \] This applies the associative property correctly by regrouping the factors. So this statement is true.

3. Second Comparison: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 - 6y \] This statement is incorrect as it uses the wrong operations (it introduces a subtraction instead of multiplying). So this statement is false.

4. Third Comparison: \[ -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \] This statement seems to equate an expression involving multiplication to one involving addition and seems incorrect. Therefore, this statement is also false.

Conclusion:

The only correct equivalent expression using the associative property is the first comparison: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y). \]

The other two comparisons are not equivalent.