To determine which option expresses the associative law of addition, let's first understand what the associative law of addition states. The associative law of addition states that the way in which numbers are grouped does not affect their sum. In other words, for any three numbers A, B, and C:
[tex]\[ (A + B) + C = A + (B + C) \][/tex]
Let's evaluate each option to see which one conforms to this law:
a)
[tex]\[ A + (B + C) \quad \text{and} \quad A + (BC) \][/tex]
This option implies the addition of A to the sum of B and C, and then compares it to adding A to the product of B and C, which is not correct according to the associative law of addition.
b)
[tex]\[ A(BC) \quad \text{and} \quad (AB) + C \][/tex]
This option mixes multiplication and addition in a way that doesn't align with the associative property of addition.
c)
[tex]\[ ABC \quad \text{and} \quad A + B + C \][/tex]
This option represents the product of A, B, and C compared to the sum of A, B, and C, which is irrelevant to the associative law of addition.
d)
[tex]\[ A + (B + C) = (A + B) + C \][/tex]
This option correctly shows that the grouping of numbers does not affect the result of their addition. Thus, it directly represents the associative law of addition.
Therefore, the option that correctly expresses the associative law of addition is:
[tex]\[ \boxed{d} \][/tex]
So, the correct answer is option d.
