To understand the given mathematical expression and determine which property it demonstrates, let's break it down step by step.
The expression given is:
[tex]\[
6 \times (7 + 3) = (6 \times 7) + (6 \times 3)
\][/tex]
This expression can be analyzed to see how it adheres to certain algebraic properties. Let's examine the left-hand side (LHS) and the right-hand side (RHS) separately.
1. Left-Hand Side (LHS):
[tex]\[
6 \times (7 + 3)
\][/tex]
According to the basic rules of arithmetic, you first perform the operation inside the parentheses (addition) and then multiply the result by 6.
[tex]\[
7 + 3 = 10
\][/tex]
Thus,
[tex]\[
6 \times 10 = 60
\][/tex]
2. Right-Hand Side (RHS):
[tex]\[
(6 \times 7) + (6 \times 3)
\][/tex]
First, perform the multiplications separately:
[tex]\[
6 \times 7 = 42 \quad \text{and} \quad 6 \times 3 = 18
\][/tex]
Then, add the two results together:
[tex]\[
42 + 18 = 60
\][/tex]
It is evident that both the LHS and the RHS yield the same value (60). This confirms that the given expression holds true.
The important point shown by this expression is how multiplication interacts with addition within a set of parentheses. This is a classic demonstration of an important property in algebra, specifically the distributive property.
Distributive Property of Multiplication over Addition:
The distributive property states that for any three whole numbers \(a\), \(b\), and \(c\),
[tex]\[
a \times (b + c) = (a \times b) + (a \times c)
\][/tex]
In this problem, \(a = 6\), \(b = 7\), and \(c = 3\). The expression \(6 \times (7 + 3)\) is equivalent to \((6 \times 7) + (6 \times 3)\), perfectly illustrating this property.
Hence, the statement shows that multiplication of whole numbers is distributive over addition.
