According to the Associative Property of Addition, the way in which numbers are grouped in addition does not affect the sum. This means that \( (b+c)+a+d \) can be grouped in various ways without changing the result.
From the given options, the expression \( d+(b+c)+a \) and the expression \( a+(b+c)+d \) can be shown to be equivalent to the original expression \( (b+c)+a+d \) due to the associative property.
Looking at the options you provided, we can simplify and assess them.
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b+(c+a)+d: This groups \(c\) and \(a\) together. It is equivalent since \( (b+c)+a+d \) can also be viewed as \( b + (c + a) + d \).
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d+(b+c)+a: This also maintains the same structure, allowing for commutativity and re-grouping accordance with the associative law.
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d + (b+c) + a: Similar to option 2, this is equivalent due to the associative property.
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a + (b+c) + d: Also valid by the associative property.
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(b+c)+d+a: This can be rearranged and stays equivalent to the original.
The correct equivalent expressions based on the associative property include:
- \( b+(c+a)+d \)
- \( d+(b+c)+a \)
- \( a+(b+c)+d \)
- \( (b+c)+d+a \)
However, since your question asked for the specific equivalent to \( (b+c)+a+d \), any of the groupings that respect the associative law are correct. The first option, b+(c+a)+d, directly reflects the associative property without changing the total value.
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