Question: Write the following expression as a single complex number.
$\frac{(1 - 2i)}{(3 - 2i)}$
- $- \frac{1}{13} - \frac{4}{13} i$
- $- \frac{1}{5} - \frac{4}{5} i$
- $\frac{7}{5} - \frac{4}{5} i$
- $\frac{7}{13} - \frac{4}{13} i$
1. **State the problem:** Simplify the expression $$\frac{1 - 2i}{3 - 2i}$$ as a single complex number.
2. **Formula and rule:** To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.
3. **Multiply numerator and denominator by the conjugate of the denominator:**
$$\frac{1 - 2i}{3 - 2i} \times \frac{3 + 2i}{3 + 2i} = \frac{(1 - 2i)(3 + 2i)}{(3 - 2i)(3 + 2i)}$$
4. **Calculate numerator:**
$$(1)(3) + (1)(2i) + (-2i)(3) + (-2i)(2i) = 3 + 2i - 6i - 4i^2$$
5. **Simplify numerator using $i^2 = -1$:**
$$3 + 2i - 6i - 4(-1) = 3 + 2i - 6i + 4 = (3 + 4) + (2i - 6i) = 7 - 4i$$
6. **Calculate denominator:**
$$(3)^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13$$
7. **Write the fraction:**
$$\frac{7 - 4i}{13} = \frac{7}{13} - \frac{4}{13}i$$
8. **Final answer:**
$$\frac{7}{13} - \frac{4}{13}i$$