1. **State the problems:** We have three expressions to simplify: - Simplify $\frac{4 - 10i}{-i}$ - Simplify $\frac{5 + 2i}{-2 - 5i}$ - Simplify $i^{19}$ 2. **Recall important rules:** - Division by a complex number can be simplified by multiplying numerator and denominator by the conjugate of the denominator. - Powers of $i$ cycle every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then repeats. --- ### Problem 1: Simplify $\frac{4 - 10i}{-i}$ 3. Multiply numerator and denominator by $i$ to remove $i$ from denominator: $$\frac{4 - 10i}{-i} \times \frac{i}{i} = \frac{(4 - 10i) i}{-i \times i}$$ 4. Calculate numerator: $$ (4)(i) - (10i)(i) = 4i - 10i^2 $$ Recall $i^2 = -1$, so: $$4i - 10(-1) = 4i + 10$$ 5. Calculate denominator: $$ -i \times i = -i^2 = -(-1) = 1$$ 6. So the expression becomes: $$\frac{10 + 4i}{1} = 10 + 4i$$ --- ### Problem 2: Simplify $\frac{5 + 2i}{-2 - 5i}$ 7. Multiply numerator and denominator by the conjugate of denominator $-2 + 5i$: $$\frac{5 + 2i}{-2 - 5i} \times \frac{-2 + 5i}{-2 + 5i} = \frac{(5 + 2i)(-2 + 5i)}{(-2 - 5i)(-2 + 5i)}$$ 8. Calculate numerator: $$5 \times (-2) + 5 \times 5i + 2i \times (-2) + 2i \times 5i = -10 + 25i - 4i + 10i^2$$ Simplify imaginary terms: $$-10 + (25i - 4i) + 10i^2 = -10 + 21i + 10(-1) = -10 + 21i - 10 = -20 + 21i$$ 9. Calculate denominator: $$(-2)^2 - (5i)^2 = 4 - 25i^2 = 4 - 25(-1) = 4 + 25 = 29$$ 10. So the expression becomes: $$\frac{-20 + 21i}{29} = -\frac{20}{29} + \frac{21}{29}i$$ --- ### Problem 3: Simplify $i^{19}$ 11. Use the cyclic property of powers of $i$: $$i^{19} = i^{4 \times 4 + 3} = (i^4)^4 \times i^3 = 1^4 \times i^3 = i^3$$ 12. Recall $i^3 = -i$ --- **Final answers:** - $\frac{4 - 10i}{-i} = 10 + 4i$ - $\frac{5 + 2i}{-2 - 5i} = -\frac{20}{29} + \frac{21}{29}i$ - $i^{19} = -i$