Let's tackle each part step-by-step and use the Distributive Property to find equivalent expressions.
### Part A
Find two different ways to calculate the area of a rectangle with dimensions 6 and 8.
1. Standard multiplication:
[tex]\[ 6 \times 8 = 48 \][/tex]
2. Distribute 6 as [tex]\(3 + 3\)[/tex] and then multiply:
[tex]\[
(3 + 3) \times 8 \\
= 3 \times 8 + 3 \times 8 \\
= 24 + 24 \\
= 48
\][/tex]
Both methods show that the area is 48 square units.
### Part B
Identify how the Distributive Property connects with given number pairs.
For clarity, rephrase the part to use the Distributive Property more effectively.
1. Distribute 6 into [tex]\( 5 + 1 \)[/tex]:
[tex]\[
6 \times 8 = (5 + 1) \times 8 \\
= 5 \times 8 + 1 \times 8 \\
= 40 + 8 \\
= 48
\][/tex]
2. Distribute 12 into [tex]\( 10 + 2 \)[/tex]:
[tex]\[
6 \times 12 = 6 \times (10 + 2) \\
= 6 \times 10 + 6 \times 2 \\
= 60 + 12 \\
= 72
\][/tex]
The idea is to break down one factor into smaller parts and apply the distributive property to calculate step-by-step.
### Part C
Use the Distributive Property:
[tex]\[
2(x + 6) = (2 \times x) + (2 \times 6) \\
So, \boxed{2x} and \boxed{2 \times 6}
\][/tex]
Thus, the equation:
[tex]\[
12 = 2(x + 6)\\
= (2x) + (12)
\][/tex]
### Part D:
Simplify the expression:
[tex]\[
(x - 2) + (x + 6)\\
Combine like terms:
= x + x + 6 - 2\\
= 2x + 4
\][/tex]
Finally, interpret [tex]\(2+6\)[/tex]:
It appears [tex]\( 2 + 6 \)[/tex] was a misunderstanding in the question, focusing on the simplification:
[tex]\[
2x + 4
\][/tex]
Summarizing the answers:
1. Part A:
[tex]\[ 6 \times 8 = 48 \][/tex]
[tex]\[ (3+3) \times 8 = 3 \times 8 + 3 \times 8 = 48 \][/tex]
2. Part B:
[tex]\[ (5+1) \times 8 = 5 \times 8 + 1 \times 8 = 48 \][/tex]
[tex]\[ 6 \times (10+2) = 6 \times 10 + 6 \times 2 = 72 \][/tex]
3. Part C:
[tex]\[ 2(x + 6) = (2x) + (2 \times 6) = 2x + 12 \][/tex]
4. Part D:
[tex]\[ (x - 2) + (x + 6) = 2x + 4 \][/tex]
