The distributive property of multiplication over addition is a fundamental property in algebra. It states that when you multiply a number by a sum, it's the same as multiplying the number by each addend and then adding the results.
Let's evaluate each of the given options to determine which one correctly represents the distributive property.
1. [tex]\( a + b = b + a \)[/tex]
This equation represents the commutative property of addition, which states that the order in which two numbers are added does not affect the sum.
2. [tex]\( a + (b + c) = (a + b) + c \)[/tex]
This equation represents the associative property of addition, which states that the grouping of numbers being added does not affect the sum.
3. [tex]\( a + c = b + c \)[/tex]
This equation does not represent a common algebraic property and is not related to the distributive property.
4. [tex]\( a(b + c) = ab + ac \)[/tex]
This equation correctly represents the distributive property. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
After examining each option, we can conclude that the fourth option:
[tex]\[ a(b + c) = ab + ac \][/tex]
is the mathematical description of the distributive property.
Therefore, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
