1. **Problem statement:** Separate the given expressions into their real and imaginary parts and write them as simple complex numbers. 2. **Recall:** A complex number is written as $a + bi$ where $a$ is the real part and $b$ is the imaginary part. --- **i) Expression:** $\frac{2 + 7i}{4 - 5i}$ 3. Multiply numerator and denominator by the conjugate of the denominator: $$\frac{2 + 7i}{4 - 5i} \times \frac{4 + 5i}{4 + 5i} = \frac{(2 + 7i)(4 + 5i)}{(4 - 5i)(4 + 5i)}$$ 4. Calculate numerator: $$(2)(4) + (2)(5i) + (7i)(4) + (7i)(5i) = 8 + 10i + 28i + 35i^2$$ 5. Since $i^2 = -1$, simplify: $$8 + 10i + 28i - 35 = (8 - 35) + (10i + 28i) = -27 + 38i$$ 6. Calculate denominator: $$(4)^2 - (5i)^2 = 16 - 25i^2 = 16 - 25(-1) = 16 + 25 = 41$$ 7. So the expression is: $$\frac{-27 + 38i}{41} = -\frac{27}{41} + \frac{38}{41}i$$ --- **ii) Expression:** $\frac{2}{2 + 3i} + \frac{1}{1 - i}$ 8. Simplify each term separately by multiplying numerator and denominator by the conjugate of the denominator. For $\frac{2}{2 + 3i}$: $$\frac{2}{2 + 3i} \times \frac{2 - 3i}{2 - 3i} = \frac{2(2 - 3i)}{(2)^2 - (3i)^2} = \frac{4 - 6i}{4 - 9(-1)} = \frac{4 - 6i}{4 + 9} = \frac{4 - 6i}{13} = \frac{4}{13} - \frac{6}{13}i$$ For $\frac{1}{1 - i}$: $$\frac{1}{1 - i} \times \frac{1 + i}{1 + i} = \frac{1 + i}{1 - i^2} = \frac{1 + i}{1 - (-1)} = \frac{1 + i}{2} = \frac{1}{2} + \frac{1}{2}i$$ 9. Add the two results: $$\left(\frac{4}{13} - \frac{6}{13}i\right) + \left(\frac{1}{2} + \frac{1}{2}i\right) = \left(\frac{4}{13} + \frac{1}{2}\right) + \left(-\frac{6}{13} + \frac{1}{2}\right)i$$ 10. Find common denominators: $$\frac{4}{13} + \frac{1}{2} = \frac{8}{26} + \frac{13}{26} = \frac{21}{26}$$ $$-\frac{6}{13} + \frac{1}{2} = -\frac{12}{26} + \frac{13}{26} = \frac{1}{26}$$ 11. Final result: $$\frac{21}{26} + \frac{1}{26}i$$ --- **iii) Expression:** $1 + i + i$ 12. Combine like terms: $$1 + i + i = 1 + 2i$$ --- **Final answers:** **i)** $-\frac{27}{41} + \frac{38}{41}i$ **ii)** $\frac{21}{26} + \frac{1}{26}i$ **iii)** $1 + 2i$