The associative property refers to the ability to regroup numbers in addition or multiplication without changing their result. Specifically:
- For addition: \( (a + b) + c = a + (b + c) \)
- For multiplication: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)
Now, let's analyze each of the given expressions:
a. \( (9 \cdot 5) + 8 + 3 \)
- You could rewrite it using the associative property for addition: \( ((9 \cdot 5) + 8) + 3 \) or \( (9 \cdot 5) + (8 + 3) \).
b. \( (5 + 8) \cdot 3 \)
- You could rewrite it using the associative property for multiplication: \( 5 + (8 \cdot 3) \) is not applicable here, but explicitly regrouping the addition isn’t used in multiplication.
c. \( (9 \cdot 5) \cdot (8 \cdot 3) \)
- This does not use the associative property directly, but you could change the grouping like: \( 9 \cdot (5 \cdot (8 \cdot 3)) \).
d. \( (5 \cdot 8) + 3 \)
- You can't apply the associative property directly since multiplication and addition can't be intermixed in this context.
Based on the above analysis, the expressions that could be rewritten using the associative property are:
a. (9 dot 5) + 8 + 3
c. (9 dot 5) dot (8 dot 3)
These expressions involve regrouping either additions or multiplications in a way consistent with the associative property.
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