15 Hardest SAT Math Questions | CollegeVine Blog - Free Printable

We are asked to simplify the complex fraction:

$$
\frac{3 - 5i}{8 + 2i}
$$

To simplify a complex fraction like this, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $8 + 2i$ is $8 - 2i$.

---

Step 1: Multiply numerator and denominator by the conjugate

$$
\frac{3 - 5i}{8 + 2i} \cdot \frac{8 - 2i}{8 - 2i}
$$

Now compute both numerator and denominator.

---

Step 2: Expand the numerator

$$
(3 - 5i)(8 - 2i) = 3 \cdot 8 + 3 \cdot (-2i) + (-5i) \cdot 8 + (-5i) \cdot (-2i)
$$

$$
= 24 - 6i - 40i + 10i^2
$$

$$
= 24 - 46i + 10i^2
$$

Recall that $i^2 = -1$, so:

$$
= 24 - 46i + 10(-1) = 24 - 46i - 10 = 14 - 46i
$$

---

Step 3: Expand the denominator

$$
(8 + 2i)(8 - 2i) = 8^2 - (2i)^2 = 64 - 4i^2
$$

$$
= 64 - 4(-1) = 64 + 4 = 68
$$

---

Step 4: Combine

$$
\frac{14 - 46i}{68} = \frac{14}{68} - \frac{46}{68}i
$$

Simplify fractions:

- $\frac{14}{68} = \frac{7}{34}$
- $\frac{46}{68} = \frac{23}{34}$

So:

$$
\frac{7}{34} - \frac{23}{34}i
$$

---

Final Answer:

$$
\boxed{\text{C)}\ \frac{7}{34} - \frac{23}{34}i}
$$

✔ Correct answer: C