Let's analyze the given problem step by step:
1. Given Sum:
[tex]\[
(4 + 5i) + (-3 + 7i)
\][/tex]
2. Correct Calculation:
You add the real parts together and the imaginary parts together:
[tex]\[
(4 + (-3)) + (5i + 7i) = 1 + 12i
\][/tex]
3. Aiko's Incorrect Rewriting:
Aiko rewrote the sum as:
[tex]\[
(-3+7) i + (4+5) i
\][/tex]
This changes both the order and the grouping of the terms, and it implies an incorrect combination of real and imaginary parts.
4. Identifying the Error:
It's clear that Aiko mistakenly combined the real parts with the imaginary parts in an incorrect manner that does not follow the standard algebraic properties used for complex numbers.
To determine exactly which property was misapplied:
- Commutative Property: This states that you can change the order of the operands in addition or multiplication without changing the result. Aiko’s error wasn’t just changing the order but changing the method of addition itself.
- Associative Property: This allows you to group numbers differently in addition or multiplication. Aiko did not just regroup; she completely altered how the real and imaginary parts were combined.
- Identity Property: The identity property involves combining with 0 (for addition) or 1 (for multiplication). No such identity element is involved in her mistake.
- Distributive Property: This property involves distributing a single operation over others, like [tex]\(a(b + c) = ab + ac\)[/tex]. Aiko's attempt to combine the real number and the coefficient of the imaginary part suggests that she misapplied the distributive property by treating it as if the terms inside the parentheses could be grouped under the [tex]\(i\)[/tex] in the same way.
Based on the detailed analysis, the correct explanation for Aiko’s mistake is:
Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part.
Therefore, the correct answer is:
Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part.
