1. **State the problem:** Simplify the expression $3\sqrt{3} + \frac{4\sqrt{2}}{2\sqrt{3}}$. 2. **Recall the rules:** - Division of radicals: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$. - Simplify fractions and radicals where possible. 3. **Simplify the fraction part:** $$\frac{4\sqrt{2}}{2\sqrt{3}} = \frac{4}{2} \times \frac{\sqrt{2}}{\sqrt{3}} = 2 \times \sqrt{\frac{2}{3}}$$ 4. **Rationalize the denominator inside the radical:** $$\sqrt{\frac{2}{3}} = \sqrt{\frac{2}{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \sqrt{\frac{2 \times 3}{3 \times 3}} = \sqrt{\frac{6}{9}} = \frac{\sqrt{6}}{3}$$ 5. **Substitute back:** $$2 \times \frac{\sqrt{6}}{3} = \frac{2\sqrt{6}}{3}$$ 6. **Rewrite the original expression:** $$3\sqrt{3} + \frac{2\sqrt{6}}{3}$$ 7. **Final answer:** The simplified form is $$3\sqrt{3} + \frac{2\sqrt{6}}{3}$$. This expression cannot be simplified further because the terms are unlike radicals.