Dadas los numeros complajos \( z_{1}=8-i \quad z_{2}=i+2 \quad z_{3}=6-3 i \quad z_{4}=-4 i \) Resolwar las opuraciones \( \begin{array}{llll}\text { a). } z_{1} \cdot z_{3} & \text { b) } z_{2}-z_{1} & \text { c) } z_{4} \cdot z_{2}-z_{3} & \text { d) } \frac{z_{3}}{z_{1}} \\ \text { e) }-\frac{z_{1}}{z_{3}} & \end{array} \)

Vamos a resolver cada una de las operaciones con los números complejos dados: 1. **\( z_{1} \cdot z_{3} \)** \[ z_{1} = 8 - i, \quad z_{3} = 6 - 3i \] \[ z_{1} \cdot z_{3} = (8 - i)(6 - 3i) \] Usamos la propiedad distributiva (o el método FOIL): \[ = 8 \cdot 6 + 8 \cdot (-3i) - i \cdot 6 - i \cdot (-3i) \] \[ = 48 - 24i - 6i + 3 \] \[ = 48 + 3 - 30i = 51 - 30i \] 2. **\( z_{2} - z_{1} \)** \[ z_{2} = i + 2, \quad z_{1} = 8 - i \] \[ z_{2} - z_{1} = (i + 2) - (8 - i) \] \[ = i + 2 - 8 + i = 2i - 6 \] 3. **\( z_{4} \cdot z_{2} - z_{3} \)** \[ z_{4} = -4i, \quad z_{2} = i + 2, \quad z_{3} = 6 - 3i \] \[ z_{4} \cdot z_{2} = (-4i)(i + 2) \] \[ = -4i^2 - 8i = 4 - 8i \quad (\text{ya que } i^2 = -1) \] Ahora restamos \( z_{3} \): \[ z_{4} \cdot z_{2} - z_{3} = (4 - 8i) - (6 - 3i) \] \[ = 4 - 8i - 6 + 3i = -2 - 5i \] 4. **\( \frac{z_{3}}{z_{1}} \)** \[ z_{3} = 6 - 3i, \quad z_{1} = 8 - i \] Para dividir números complejos, multiplicamos por el conjugado del denominador: \[ \frac{z_{3}}{z_{1}} = \frac{6 - 3i}{8 - i} \cdot \frac{8 + i}{8 + i} \] \[ = \frac{(6 - 3i)(8 + i)}{(8 - i)(8 + i)} \] Calculamos el denominador: \[ (8 - i)(8 + i) = 64 + 1 = 65 \] Ahora el numerador: \[ (6 - 3i)(8 + i) = 48 + 6i - 24i - 3i^2 = 48 - 18i + 3 = 51 - 18i \] Entonces: \[ \frac{z_{3}}{z_{1}} = \frac{51 - 18i}{65} = \frac{51}{65} - \frac{18}{65}i \] 5. **\( -\frac{z_{1}}{z_{3}} \)** \[ z_{1} = 8 - i, \quad z_{3} = 6 - 3i \] Similar al caso anterior: \[ -\frac{z_{1}}{z_{3}} = -\frac{8 - i}{6 - 3i} \cdot \frac{6 + 3i}{6 + 3i} \] Denominador: \[ (6 - 3i)(6 + 3i) = 36 + 9 = 45 \] Numerador: \[ (8 - i)(6 + 3i) = 48 + 24i - 6i - 3i^2 = 48 + 18i + 3 = 51 + 18i \] Entonces