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 mathematical property that she made an error using?
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.

Answer 1

Consider the expression below.

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

Aiko rewrites the equation by factoring out the i's from the expression and wrote it as,

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

But this is wrong because 'i' is an imaginary number which is with '5' and '7'. So, she cannot factor out the 'i' because it is not present in both '-3' and '4'.

She could rewrote it as $(4-3)+i(5+7)$

So, Aiko incorrectly used the distributive property by combining the real number and the coefficient of the imaginary part.

So, Option 4 is the correct answer.

Answer 2

In this item, it can be seen that from an expression of,
                                  (4 + 5i) + (-3 + 7i)
Aiko transformed the equation by factoring out the i's from the expression and wrote it as,
                                   (4 + 5)i + (-3 + 7)i
Now this is wrong because i there is not a variable but a notation that 5 and 7 are imaginary numbers. In fact, she cannot factor out the i's because it is not present in both -3 and 4. The answer therefore to this item is the last sentence, "Aiko incorrectly used the distributive property by combining ..."