To determine which equation demonstrates the additive identity property, we first need to understand what the additive identity property is. The additive identity property states that adding zero to any number leaves it unchanged.
Let's analyze each of the given equations to see which one fits this property:
1. [tex]\((7 + 4i) + (7 - 4i) = 14\)[/tex]
This equation involves adding the complex number [tex]\(7 + 4i\)[/tex] to [tex]\(7 - 4i\)[/tex]. The imaginary parts cancel each other out, and the real parts sum to [tex]\(14\)[/tex]. This equation does not demonstrate the additive identity property, as it does not involve adding zero.
2. [tex]\((7 + 4i) + 0 = 7 + 4i\)[/tex]
This equation shows the complex number [tex]\(7 + 4i\)[/tex] being added to zero, and the result is the same number [tex]\(7 + 4i\)[/tex]. This is precisely what the additive identity property states: adding zero to a number leaves it unchanged. Therefore, this equation demonstrates the additive identity property.
3. [tex]\((7 + 4i)(1) - 7 + 4i\)[/tex]
This equation involves multiplying the complex number [tex]\(7 + 4i\)[/tex] by [tex]\(1\)[/tex], which leaves it unchanged, and then performing operations including subtraction and addition. This equation involves more than just the additive identity property, so it does not demonstrate it.
4. [tex]\((7 + 4i) + (-7 - 4i) = 0\)[/tex]
This equation shows the complex number [tex]\(7 + 4i\)[/tex] being added to its additive inverse [tex]\(-7 - 4i\)[/tex], resulting in zero. This demonstrates the concept of additive inverses, not the additive identity property.
Given these evaluations, the equation that clearly demonstrates the additive identity property is:
[tex]\[
(7 + 4i) + 0 = 7 + 4i
\][/tex]
Therefore, the correct equation demonstrating the additive identity property is:
[tex]\[
2
\][/tex]
