Question

According to the Associative Property of Addition, which of the following is equivalent to the expression (b+c)+a+d(b+c)+a+d ?(1 point) Responses b+(c+a)+db+(c+a)+d b plus left parenthesis c plus a right parenthesis plus d d+(b+c)+ad+(b+c)+a d plus left parenthesis b plus c right parenthesis plus a d+a+(b+c)d+a+(b+c) d plus a plus left parenthesis b plus c right parenthesis (b+c)+d+a

Answer 1

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.

1. **b+(c+a)+d**: This groups \(c\) and \(a\) together. It is equivalent since \( (b+c)+a+d \) can also be viewed as \( b + (c + a) + d \).

2. **d+(b+c)+a**: This also maintains the same structure, allowing for commutativity and re-grouping accordance with the associative law.

3. **d + (b+c) + a**: Similar to option 2, this is equivalent due to the associative property.

4. **a + (b+c) + d**: Also valid by the associative property.

5. **(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.