Question

which of the following complex numbers is equivalent to \\(\frac{3 - 5i}{8 + 2i}\\)? (note: \\(i = \sqrt{-1}\\))

a) \\(\frac{3}{8} - \frac{5i}{2}\\)

b) \\(\frac{3}{8} + \frac{5i}{2}\\)

c) \\(\frac{7}{34} - \frac{23i}{34}\\)

d) \\(\frac{7}{34} + \frac{23i}{34}\\)

Explanation:

Multiply by the complex conjugate of the denominator

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

Expand the numerator and denominator

$$ \text{Numerator: } (3 - 5i)(8 - 2i) = 24 - 6i - 40i + 10i^2 = 24 - 46i - 10 = 14 - 46i $$

$$ \text{Denominator: } (8 + 2i)(8 - 2i) = 8^2 - (2i)^2 = 64 - (-4) = 68 $$

Simplify the resulting fraction

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

Answer:

• (A) \(\frac{3}{8} - \frac{5i}{2}\)
• (B) \(\frac{3}{8} + \frac{5i}{2}\)
• (C) \(\frac{7}{34} - \frac{23i}{34}\) (Correct answer)
• (D) \(\frac{7}{34} + \frac{23i}{34}\)