Final answer:
To simplify the complex expression (1-2i)/(3-2i), we multiply numerator and denominator by the complex conjugate (3+2i), which results in the simplification to a single complex number 7/13 - 4/13i. However, a typo in the question options suggests the correct answer as (-1/13 - 4/13i) instead of (7/13 - 4/13i). option d is the correct answer.
Step-by-step explanation:
The question asks us to simplify the complex expression 1-2i/3-2i. To express it as a single complex number, we can use a method called complex conjugation. The complex conjugate of the denominator (3-2i) is 3+2i. We multiply both the numerator and the denominator by this conjugate.
The multiplication process is as follows:
(1 - 2i)(3 + 2i) / (3 - 2i)(3 + 2i)
This simplifies to:
(3 + 2i - 6i - 4i²) / (9 + 6i - 6i - 4i²)
Recognizing that i² = -1, we continue simplifying:
(3 - 4i² - 4i) / (9 - 4i²)
Which further simplifies to:
(3 + 4 - 4i) / (9 + 4)
Thus, the single complex number would be:
7/13 - 4/13i
So the correct answer is c. -1/13-4/13i, but due to the positive results of real part calculation, the answer should have been d. 7/13 - 4/13i instead.