To determine which expressions are equivalent using the Associative Property, we first need to understand what the Associative Property is. The Associative Property states that the way in which numbers are grouped in multiplication (or addition) does not change their product (or sum). For multiplication, it can be expressed as:
\[ a \cdot (b \cdot c) = (a \cdot b) \cdot c \]
Now, let's analyze the expressions given in the question one by one.
-
First Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \]
This expression is applying the associative property to rearrange the grouping of the terms. This is valid under the associative property, so this expression is equivalent.
-
Second Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 - 6y \]
This expression is incorrect as it applies subtraction after multiplication, which does not follow the associative property. So, this expression is not equivalent.
-
Third Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \]
This appears to be an equation aiming to simplify or evaluate a multiplication operation, which does not follow the associative property concept. Therefore, this expression is not equivalent.
-
Fourth Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \]
Similar to the third expression, this does not adhere to the associative property and suggests a simplification that may or may not be accurate. Therefore, this expression is not equivalent.
Conclusion
The only expression that uses the Associative Property correctly to show equivalence is the first expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \]