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Bot GPT 4-o mini The Associative Property states that the way in which numbers are grouped in an expression does not change their sum or product. In mathematical terms, for any numbers \( a, b, \) and \( c \):
- For addition: \( (a + b) + c = a + (b + c) \)
- For multiplication: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)
To determine which expressions are equivalent, let's analyze each option you provided:
1. **Expression 1:**
\[
-3 - (4x + 2) * (-6)g = 18 - 8z
\]
There's no clear application of the associative property here; we need more context to assess equivalence since both expressions don't seem to connect.
2. **Expression 2:**
\[
-3 - (4x + 2) + 6y = -7xy
\]
Similar to the first, it's hard to see whether theyβre equivalent without more context.
3. **Expression 3:**
\[
-3 - (4x + 2) * (-6)y = (-3 * 4x)(-2 + 6y)
\]
Looking at this, we can use the distributive property but not really confirm equivalence unless we simplify both sides (which might involve non-associative operations).
4. **Expression 4:**
\[
-3(4x + 2) * (-6)y = (-3 * 4x)(-2) - 6y
\]
This expression can be manipulated using the distributive property. The left side expands to \( -3 * (-6) * (4x + 2)y \). On the right side, after distributing \( (-3 * 4x)(-2) + ...\), it might yield something associative but again needs further checking.
To summarize, without applying specific values or additional algebraic techniques, it's challenging to explicitly determine viable equivalences using associativity alone for all expressions.
You might find some expressions simplify to yield transformation into others through grouping or distribution, but this requires testing simplified forms or substituting values to check equivalency.
In essence, without further simplification, we can't determine which expressions equate based solely on the information provided. Would you like to focus on simplifying them further, or do you have particular substitutions in mind?