Final answer:
The expression 3 - 4i / 1 - 2i, when simplified by multiplying both numerator and denominator by the complex conjugate of the denominator, yields a result of (11/5) - (2/5)i, which corresponds to option D.
Step-by-step explanation:
To simplify the expression 3 - 4i / 1 - 2i, we must multiply both the numerator and the denominator by the complex conjugate of the denominator to remove the imaginary unit i from the denominator. The complex conjugate of 1 - 2i is 1 + 2i. By multiplying the numerator and denominator by this complex conjugate, we get:
(3 - 4i)(1 + 2i) / (1 - 2i)(1 + 2i)
This simplifies to:
(3 + 6i - 4i - 8i²) / (1 + 2i - 2i - 4i²)
Since i² = -1, we can further simplify this to:
(3 + 2i + 8) / (1 + 4)
Which simplifies to:
(11 + 2i) / 5
Dividing both terms in the numerator by 5 gives us the answer:
(11/5) + (2/5)i
Therefore, the simplified expression is option D: 11/5 - 2/5i.