To determine which expressions are equivalent using the Associative Property, let’s first understand what the Associative Property states. It refers to the way in which numbers are grouped in the operation; for multiplication, \( a \cdot (b \cdot c) = (a \cdot b) \cdot c\).
Now, let's evaluate the given expressions one by one:
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First Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \] This expression incorrectly groups terms and doesn't seem to simplify correctly.
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Second Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot (-2 \cdot -6y) \] This expression is valid under the Associative property. It correctly rearranges the grouping of numbers.
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Third Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \] This expression seems to suggest that the left side evaluates to \(-7xy\). However, we can calculate it to see if it's true:
- Evaluate \( 4x \cdot -2 = -8x \).
- Then calculate \( -3 \cdot (-8x) \cdot -6y = -3 \cdot -8 \cdot -6xy = 144xy\). Hence, it's not equivalent to \(-7xy\).
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Fourth Expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \] Trying to rearrange would lead to another result and hints it doesn't combine to that value. Thus not equivalent.
Final Assessment: Based on the evaluations, the only expression that correctly reflects the Associative Property for equivalence in the context is:
- Correct Equivalent Expression: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)
This is the only expression that accurately reflects the Associative Property as it rearranges groupings without changing the overall product.
3 answers