To determine which property is illustrated by the statement [tex]\(a \times (b + c) = a \times b + a \times c\)[/tex], let’s break down the equation and analyze it step by step.
1. Analyze the Left-Hand Side (LHS):
[tex]\[
a \times (b + c)
\][/tex]
Here, [tex]\(a\)[/tex] is being multiplied by the entire sum [tex]\(b + c\)[/tex].
2. Analyze the Right-Hand Side (RHS):
[tex]\[
a \times b + a \times c
\][/tex]
Here, [tex]\(a\)[/tex] is multiplied by [tex]\(b\)[/tex] and [tex]\(a\)[/tex] is separately multiplied by [tex]\(c\)[/tex], and then the two products are added together.
3. Compare LHS and RHS:
The left-hand side shows [tex]\(a\)[/tex] being distributed across the sum of [tex]\(b\)[/tex] and [tex]\(c\)[/tex]. In the right-hand side, [tex]\(a\)[/tex] is distributed and multiplied individually by [tex]\(b\)[/tex] and [tex]\(c\)[/tex], then the results are summed.
This pattern of distribution indicates a property associated with distributing a multiplication operation over addition. Therefore, the property being illustrated is:
Distributive Property
The distributive property states that:
[tex]\[
a \times (b + c) = a \times b + a \times c
\][/tex]
Option d. Distributive is the correct answer.
Hence, the property illustrated by the statement [tex]\(a \times (b + c) = a \times b + a \times c\)[/tex] is the Distributive property.
