1. Problem: Simplify each complex fraction to its simplest form. 2. Simplify (3 - 6i)/3i: - Multiply numerator and denominator by the conjugate of the denominator's imaginary part, which is -3i, to remove the imaginary unit from the denominator: $$\frac{3-6i}{3i} \times \frac{-i}{-i} = \frac{(3-6i)(-i)}{3i(-i)}$$ - Calculate numerator: $$(3)(-i) - (6i)(-i) = -3i + 6i^2 = -3i + 6(-1) = -3i - 6$$ - Calculate denominator: $$3i \times -i = -3i^2 = -3(-1) = 3$$ - Now fraction is: $$\frac{-3i - 6}{3} = \frac{-6}{3} + \frac{-3i}{3} = -2 - i$$ 3. Simplify 50/(4 - 3i): - Multiply numerator and denominator by the conjugate of the denominator: $$\frac{50}{4-3i} \times \frac{4+3i}{4+3i} = \frac{50(4+3i)}{(4)^2 - (3i)^2}$$ - Calculate denominator: $$16 - 9i^2 = 16 - 9(-1) = 16 + 9 = 25$$ - Calculate numerator: $$50 \times 4 + 50 \times 3i = 200 + 150i$$ - Final simplified form: $$\frac{200 + 150i}{25} = 8 + 6i$$ 4. Simplify (2 - i)/(3 - i): - Multiply numerator and denominator by the conjugate of the denominator: $$\frac{2 - i}{3 - i} \times \frac{3 + i}{3 + i} = \frac{(2 - i)(3 + i)}{(3)^2 - (i)^2}$$ - Calculate denominator: $$9 - (-1) = 10$$ - Calculate numerator: $$(2)(3) + (2)(i) - (i)(3) - (i)(i) = 6 + 2i - 3i - (-1) = 6 - i + 1 = 7 - i$$ - Final simplified form: $$\frac{7 - i}{10} = \frac{7}{10} - \frac{1}{10}i$$ Answer: (a) $-2 - i$ (b) $8 + 6i$ (c) $\frac{7}{10} - \frac{1}{10}i$