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Bot GPT 4-o mini To determine which expressions are equivalent using the Associative Property of multiplication, let's recall that the Associative Property states that when multiplying three or more numbers together, the way in which the numbers are grouped does not change the product. This can be represented as:
\[
(a \cdot b) \cdot c = a \cdot (b \cdot c)
\]
Now let's analyze the expressions given:
1. **Expression 1:** \(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)
2. **Expression 2:** \(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\)
3. **Expression 3:** \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot (-2 \cdot -6y)\)
4. **Expression 4:** \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y\)
### Assessing the Expressions
**Expressions 1 and 2:**
These expressions state equalities that compare \(-3 \cdot (4x \cdot -2) \cdot -6y\) with two different values: \(18 - 8x\) and \(-7xy\). We would need to calculate \(-3 \cdot (4x \cdot -2) \cdot -6y\) and see if it is equivalent to either expression. However, without calculation, we cannot directly verify these two are equivalent to the original expression simply under the associative property.
**Expression 3:**
This expression states that \(-3 \cdot (4x \cdot -2) \cdot -6y\) can be regrouped as \((-3 \cdot 4x) \cdot (-2 \cdot -6y)\). This employs the associative property correctly.
**Expression 4:**
This expression also uses the associative property. It can be interpreted correctly as:\((-3 \cdot 4x) \cdot -2 \cdot -6y\). Despite being different in structure, it follows the associative property.
### Conclusion
Expressions 3 and 4 are equivalent forms of the original expression due to the application of the Associative Property of multiplication.
**In summary:**
- Expression 3: Equivalent
- Expression 4: Equivalent
Expressions 1 and 2 propose equations that cannot be verified without evaluation but may or may not be equivalent depending on the specific values for \(x\) and \(y\).