1. **Problem Statement:** Simplify each complex number expression step-by-step. 2. **Recall:** For complex numbers, addition/subtraction is done by combining real parts and imaginary parts separately. Multiplication uses distributive property and $i^2 = -1$. 3. **Simplify each expression:** - $-3 + 6i - (3 - 8i) = -3 + 6i - 3 + 8i = (-3 - 3) + (6i + 8i) = -6 + 14i$ - $-3i - 3 - 8i - 8i = -3 - (3i + 8i + 8i) = -3 - 19i$ - $-2i - 6i(-3 + 8i) = -2i - 6i \times (-3) + 6i \times 8i = -2i + 18i + 48i^2 = (-2i + 18i) + 48(-1) = 16i - 48$ - $(-4 - 5i)(3 + 7i) = (-4)(3) + (-4)(7i) + (-5i)(3) + (-5i)(7i) = -12 - 28i - 15i - 35i^2 = -12 - 43i + 35 = 23 - 43i$ - $-2i(5 + 6i)(-1 - 7i)$ First multiply $(5 + 6i)(-1 - 7i)$: $= 5 \times (-1) + 5 \times (-7i) + 6i \times (-1) + 6i \times (-7i) = -5 - 35i - 6i - 42i^2 = -5 - 41i + 42 = 37 - 41i$ Now multiply by $-2i$: $-2i(37 - 41i) = -2i \times 37 + (-2i) \times (-41i) = -74i + 82i^2 = -74i + 82(-1) = -74i - 82$ - $\frac{10}{6i}$ Multiply numerator and denominator by $-i$ to rationalize: $\frac{10}{6i} \times \frac{-i}{-i} = \frac{-10i}{6i \times -i} = \frac{-10i}{6i^2} = \frac{-10i}{6(-1)} = \frac{-10i}{-6} = \frac{10i}{6} = \frac{5i}{3}$ - $4 - 10i$ Already simplified. - $-i$ Already simplified. - $-2 - i$ Already simplified. - $4 + 10i$ Already simplified. - $5 + 2i$ Already simplified. - $-2 - 5i$ Already simplified. - $-3(4 + 4i) - 5(8 + 5i)$ Calculate each term: $-3(4 + 4i) = -12 - 12i$ $-5(8 + 5i) = -40 - 25i$ Sum: $(-12 - 12i) + (-40 - 25i) = (-12 - 40) + (-12i - 25i) = -52 - 37i$ - $i^{19}$ Recall $i^4 = 1$, so $i^{19} = i^{4 \times 4 + 3} = (i^4)^4 \times i^3 = 1^4 \times i^3 = i^3 = i^2 \times i = (-1) \times i = -i$ - $i^{64}$ Since $i^4 = 1$, $i^{64} = (i^4)^{16} = 1^{16} = 1$ **Final answers:** 1. $-6 + 14i$ 2. $-3 - 19i$ 3. $-48 + 16i$ 4. $23 - 43i$ 5. $-82 - 74i$ 6. $\frac{5i}{3}$ 7. $4 - 10i$ 8. $-i$ 9. $-2 - i$ 10. $4 + 10i$ 11. $5 + 2i$ 12. $-2 - 5i$ 13. $-52 - 37i$ 14. $-i$ 15. $1$