1. **State the problem:** Simplify the expression $$\frac{3\sqrt{2} - 2\sqrt{3}i}{3\sqrt{2} + 2\sqrt{3}i}$$ where $i$ is the imaginary unit. 2. **Formula and rule:** To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $3\sqrt{2} + 2\sqrt{3}i$ is $3\sqrt{2} - 2\sqrt{3}i$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{3\sqrt{2} - 2\sqrt{3}i}{3\sqrt{2} + 2\sqrt{3}i} \times \frac{3\sqrt{2} - 2\sqrt{3}i}{3\sqrt{2} - 2\sqrt{3}i} = \frac{(3\sqrt{2} - 2\sqrt{3}i)^2}{(3\sqrt{2})^2 - (2\sqrt{3}i)^2}$$ 4. **Calculate denominator:** $$(3\sqrt{2})^2 = 9 \times 2 = 18$$ $$(2\sqrt{3}i)^2 = 4 \times 3 \times i^2 = 12 \times (-1) = -12$$ So denominator is $$18 - (-12) = 18 + 12 = 30$$ 5. **Calculate numerator:** Expand $$(3\sqrt{2} - 2\sqrt{3}i)^2 = (3\sqrt{2})^2 - 2 \times 3\sqrt{2} \times 2\sqrt{3}i + (2\sqrt{3}i)^2$$ Calculate each term: $$(3\sqrt{2})^2 = 18$$ $$-2 \times 3\sqrt{2} \times 2\sqrt{3}i = -12\sqrt{6}i$$ $$(2\sqrt{3}i)^2 = -12$$ Sum these: $$18 - 12\sqrt{6}i - 12 = 6 - 12\sqrt{6}i$$ 6. **Write the simplified expression:** $$\frac{6 - 12\sqrt{6}i}{30} = \frac{6}{30} - \frac{12\sqrt{6}}{30}i = \frac{1}{5} - \frac{2\sqrt{6}}{5}i$$ **Final answer:** $$\boxed{\frac{1}{5} - \frac{2\sqrt{6}}{5}i}$$