1. **State the problem:** We need to multiply the complex numbers and simplify powers of the imaginary unit $i$. 2. **Recall formulas and rules:** - Multiplying complex numbers: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$ - Powers of $i$: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and powers repeat every 4. 3. **Multiply $(9 - 6i)(7 - 3i)$:** $$ (9)(7) - (6)(-3) + ((9)(-3) + (-6)(7))i = 63 + 18 + (-27 - 42)i = 81 - 69i $$ 4. **Multiply $(12 + 5i)(10 + 13i)$:** $$ (12)(10) - (5)(13) + ((12)(13) + (5)(10))i = 120 - 65 + (156 + 50)i = 55 + 206i $$ 5. **Simplify powers of $i$ using modulo 4:** - $i^8 = (i^4)^2 = 1^2 = 1$ - $i^{800} = (i^4)^{200} = 1^{200} = 1$ - $i^6 = i^{4+2} = i^4 imes i^2 = 1 imes (-1) = -1$ - $i^{560} = (i^4)^{140} = 1^{140} = 1$ - $i^{16} = (i^4)^4 = 1^4 = 1$ **Final answers:** - $(9 - 6i)(7 - 3i) = 81 - 69i$ - $(12 + 5i)(10 + 13i) = 55 + 206i$ - $i^8 = 1$ - $i^{800} = 1$ - $i^6 = -1$ - $i^{560} = 1$ - $i^{16} = 1$