1. Problem (g): Simplify $ (3 - 4i) + (6 + i) $. Step 1: Add the real parts: $3 + 6 = 9$. Step 2: Add the imaginary parts: $-4i + i = -3i$. Step 3: Combine results: $9 - 3i$. Answer: $9 - 3i$. 2. Problem (h): Simplify $ (\sqrt{9} - 4i) - (-3 + \sqrt{-16}) $. Step 1: Calculate $\sqrt{9} = 3$. Step 2: Calculate $\sqrt{-16} = 4i$. Step 3: Substitute: $(3 - 4i) - (-3 + 4i)$. Step 4: Distribute minus: $3 - 4i + 3 - 4i$. Step 5: Add real parts: $3 + 3 = 6$. Step 6: Add imaginary parts: $-4i - 4i = -8i$. Step 7: Combine results: $6 - 8i$. Answer: $6 - 8i$. 3. Problem (i): Simplify $ (3 - 2i)(2 - 4i) $. Step 1: Use distributive property: $$ (3)(2) + (3)(-4i) + (-2i)(2) + (-2i)(-4i) $$ Step 2: Calculate each term: $6 - 12i - 4i + 8i^2$ Step 3: Recall $i^2 = -1$, so $8i^2 = 8(-1) = -8$. Step 4: Combine like terms: Real: $6 - 8 = -2$ Imaginary: $-12i - 4i = -16i$ Step 5: Final result: $-2 - 16i$. Answer: $-2 - 16i$. 4. Problem 2: Find $x$ and $y$ such that $$ \frac{(3 - i)(3 + i)}{2 + 2i} = x + yi $$ Step 1: Calculate numerator: $$ (3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 10 $$ Step 2: Denominator is $2 + 2i$. Step 3: Multiply numerator and denominator by conjugate of denominator $2 - 2i$: $$ \frac{10}{2 + 2i} \times \frac{2 - 2i}{2 - 2i} = \frac{10(2 - 2i)}{(2 + 2i)(2 - 2i)} $$ Step 4: Calculate denominator: $$ 2^2 - (2i)^2 = 4 - (-4) = 8 $$ Step 5: Calculate numerator: $$ 10(2 - 2i) = 20 - 20i $$ Step 6: Divide numerator by denominator: $$ \frac{20}{8} - \frac{20i}{8} = \frac{5}{2} - \frac{5}{2}i $$ Answer: $x = \frac{5}{2}$, $y = -\frac{5}{2}$. 5. Problem 3 (c): Simplify $ \frac{4 - 8i}{2i} $. Step 1: Multiply numerator and denominator by $-i$ to remove $i$ from denominator: $$ \frac{4 - 8i}{2i} \times \frac{-i}{-i} = \frac{(4 - 8i)(-i)}{2i(-i)} $$ Step 2: Calculate numerator: $$ 4(-i) - 8i(-i) = -4i + 8i^2 = -4i + 8(-1) = -4i - 8 $$ Step 3: Calculate denominator: $$ 2i(-i) = 2(-i^2) = 2(1) = 2 $$ Step 4: Divide numerator by denominator: $$ \frac{-8 - 4i}{2} = -4 - 2i $$ Answer: $-4 - 2i$. 6. Problem 3 (b): Simplify $ \frac{13}{2 - 3i} $. Step 1: Multiply numerator and denominator by conjugate $2 + 3i$: $$ \frac{13}{2 - 3i} \times \frac{2 + 3i}{2 + 3i} = \frac{13(2 + 3i)}{(2)^2 + (3)^2} = \frac{13(2 + 3i)}{4 + 9} = \frac{13(2 + 3i)}{13} $$ Step 2: Simplify numerator and denominator: $$ 2 + 3i $$ Answer: $2 + 3i$. 7. Problem 3 (c): Simplify $ \frac{5 - i}{3 - i} $. Step 1: Multiply numerator and denominator by conjugate $3 + i$: $$ \frac{5 - i}{3 - i} \times \frac{3 + i}{3 + i} = \frac{(5 - i)(3 + i)}{3^2 + 1^2} = \frac{(5)(3) + (5)(i) - (i)(3) - (i)(i)}{9 + 1} $$ Step 2: Calculate numerator: $$ 15 + 5i - 3i - i^2 = 15 + 2i - (-1) = 15 + 2i + 1 = 16 + 2i $$ Step 3: Denominator: $$ 10 $$ Step 4: Final result: $$ \frac{16}{10} + \frac{2}{10}i = \frac{8}{5} + \frac{1}{5}i $$ Answer: $\frac{8}{5} + \frac{1}{5}i$. 8. Problem 4: Simplify $ (1 - i)^{10} $. Step 1: Express $1 - i$ in polar form: $$ r = \sqrt{1^2 + (-1)^2} = \sqrt{2} $$ $$ \theta = \arctan\left(\frac{-1}{1}\right) = -\frac{\pi}{4} $$ Step 2: Use De Moivre's theorem: $$ (r e^{i\theta})^{10} = r^{10} e^{i 10 \theta} = (\sqrt{2})^{10} e^{i 10 (-\pi/4)} = (2^{5}) e^{-i \frac{10\pi}{4}} = 32 e^{-i \frac{5\pi}{2}} $$ Step 3: Simplify angle: $$ e^{-i \frac{5\pi}{2}} = e^{-i (2\pi + \frac{\pi}{2})} = e^{-i 2\pi} e^{-i \frac{\pi}{2}} = 1 \times e^{-i \frac{\pi}{2}} $$ Step 4: Convert back to rectangular form: $$ 32 (\cos(-\frac{\pi}{2}) + i \sin(-\frac{\pi}{2})) = 32 (0 - i) = -32i $$ Answer: $-32i$. 9. Problem 5: Simplify $ \left(1 + \frac{2i}{3}\right) \left(2 + 3i5 + 4i6\right) $ in form $x + yi$. Note: Assuming $3i5 = 3 \times i \times 5 = 15i$ and $4i6 = 4 \times i \times 6 = 24i$. Step 1: Simplify inside second parentheses: $$ 2 + 15i + 24i = 2 + 39i $$ Step 2: First parentheses: $$ 1 + \frac{2i}{3} = 1 + \frac{2}{3}i $$ Step 3: Multiply: $$ (1)(2 + 39i) + \frac{2}{3}i (2 + 39i) = 2 + 39i + \frac{4}{3}i + \frac{78}{3}i^2 $$ Step 4: Simplify $i^2 = -1$: $$ 2 + 39i + \frac{4}{3}i - 26 $$ Step 5: Combine real parts: $$ 2 - 26 = -24 $$ Step 6: Combine imaginary parts: $$ 39i + \frac{4}{3}i = \frac{117}{3}i + \frac{4}{3}i = \frac{121}{3}i $$ Answer: $-24 + \frac{121}{3}i$. 10. Problem 6: Determine quadrant of angles: (a) 72°: Quadrant I (0° to 90°) (b) 215°: Quadrant III (180° to 270°) (c) 135°: Quadrant II (90° to 180°) (d) 340°: Quadrant IV (270° to 360°) 11. Problem 7: Find negative measure of angles: (a) 85°: $85° - 360° = -275°$ (b) 155°: $155° - 360° = -205°$ (c) 317°: $317° - 360° = -43°$ (d) 249°: $249° - 360° = -111°$ 12. Problem 8: Find two angles (positive and negative) with same terminal side: (c) 160°: Positive: 160°, Negative: $160° - 360° = -200°$ (b) -35°: Positive: $-35° + 360° = 325°$, Negative: -35° (c) -240°: Positive: $-240° + 360° = 120°$, Negative: -240° (d) 249°: Positive: 249°, Negative: $249° - 360° = -111°$