The problem in the image requires identifying the error a student made while dividing complex numbers and then choosing the correct answer from the multiple-choice options.

Let’s break it down:

Analysis of the Student's Error

In complex division, the correct approach is to multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate reverses the sign of the imaginary part, which helps to eliminate imaginary terms in the denominator (yielding a real number).

It seems from the student’s work that they multiplied both the numerator and denominator by their own complex conjugates rather than by the conjugate of the denominator only.

Correct Answer

Based on this analysis:

• Option C is correct: "The student multiplied the numerator and denominator each by their own complex conjugates rather than multiplying both by the complex conjugate of the denominator."

This choice correctly describes the error made by the student in the calculation.

Would you like a detailed breakdown on how to correctly divide complex numbers by using the complex conjugate?

Here are 5 related questions to further your understanding:

1. How do you find the complex conjugate of a complex number?
2. Why is it important to use the complex conjugate when dividing complex numbers?
3. What happens if you multiply by the numerator’s conjugate instead of the denominator’s?
4. How can you simplify complex fractions after multiplying by the conjugate?
5. How do you simplify expressions with imaginary numbers?

Tip: Always remember to multiply by the conjugate of the denominator to eliminate the imaginary part when dividing complex numbers.