To determine which of the given expressions demonstrates the distributive property, let's examine each option closely for the specific property.
The distributive property states that [tex]\( a(b + c) = ab + ac \)[/tex].
### Option 1: [tex]\( 3 + 4 + 5 = 4 + 3 + 5 \)[/tex]
This expression is simply rearranging the numbers within addition, showing the commutative property of addition, which states that [tex]\( a + b = b + a \)[/tex]. Here, it is demonstrating that the sum remains the same irrespective of the order. This is not illustrating the distributive property.
### Option 2: [tex]\( -2(5 + 7) = -2(7 + 5) \)[/tex]
This expression is also showcasing the commutative property, but within the parentheses. It shows that the addition inside the parentheses can be rearranged without changing the result, i.e., [tex]\( 5 + 7 = 7 + 5 \)[/tex]. This does not demonstrate the distributive property.
### Option 3: [tex]\( 3(-8 + 1) = 3(-8) + 3(1) \)[/tex]
This expression is an example of the distributive property. Here’s why:
- According to the distributive property, [tex]\( a(b + c) = ab + ac \)[/tex]
- Let’s assign values: [tex]\( a = 3 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 1 \)[/tex]
Now, applying the distributive property:
[tex]\[ 3(-8 + 1) = 3(-8) + 3(1) \][/tex]
Breaking it down:
- On the left side: [tex]\( 3 \times (-8 + 1) = 3 \times (-7) = -21 \)[/tex]
- On the right side: [tex]\( 3(-8) + 3(1) = -24 + 3 = -21 \)[/tex]
Both sides are equal, which confirms it perfectly illustrates the distributive property.
### Option 4: [tex]\( 6[(7)(-2)] = [(6)(7)](-2) \)[/tex]
This expression is using associative property of multiplication, which allows the grouping of numbers in a multiplication equation to change without affecting the result, indicating [tex]\( (ab)c = a(bc) \)[/tex]. Here:
[tex]\[ 6 \cdot (7 \cdot -2) = (6 \cdot 7) \cdot -2 \][/tex]
While both sides will give the same result, this does not demonstrate the distributive property as stated earlier.
### Conclusion
The expression that correctly demonstrates the distributive property is:
[tex]\[ 3(-8 + 1) = 3(-8) + 3(1) \][/tex]
