Question

aiko is finding the sum \\((4 + 5i) + (-3 + 7i)\\). she rewrites the sum as \\((-3 + 7)i + (4 + 5)i\\). which statement explains the error aiko made by using a mathematical property incorrectly?

aiko incorrectly used the commutative property by changing the order of the two complex numbers
aiko incorrectly used the associative property by changing the order of the two complex numbers
aiko incorrectly used the identity property by combining the real number and the coefficient of the imaginary part
aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part

Explanation:

Analyze the original expression and Aiko's rewrite

Original expression:

$$ (4 + 5i) + (-3 + 7i) $$

Aiko's rewritten expression:

$$ (-3 + 7)i + (4 + 5)i $$

Identify the mathematical error

The original terms are complex numbers of the form \(a + bi\).
Aiko factored out \(i\) incorrectly from the entire expressions, changing the real parts \(4\) and \(-3\) into coefficients of \(i\).
This incorrectly applies the distributive property by grouping the real parts and the imaginary coefficients together under a single factor of \(i\).

Match with the correct option

The fourth option states: "Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part."
This accurately describes the error of distributing \(i\) to the real parts \(4\) and \(-3\) where it does not belong.

Answer:

• (A) Aiko incorrectly used the commutative property by changing the order of the two complex numbers
• (B) Aiko incorrectly used the associative property by changing the order of the two complex numbers
• (C) Aiko incorrectly used the identity property by combining the real number and the coefficient of the imaginary part
• (D) Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part (Correct answer)