1. **State the problem:** Rationalize the expression $$\frac{3\sqrt{3}+\sqrt{5}}{3\sqrt{3}-\sqrt{5}} \times \frac{3\sqrt{3}+\sqrt{5}}{3\sqrt{3}+\sqrt{5}}$$ 2. **Understand the goal:** Rationalizing the denominator means to eliminate the surds (square roots) from the denominator. 3. **Multiply numerator and denominator by the conjugate:** The conjugate of the denominator $$3\sqrt{3}-\sqrt{5}$$ is $$3\sqrt{3}+\sqrt{5}$$. Multiplying by this conjugate will help remove the surds in the denominator. 4. **Write the expression:** $$\frac{3\sqrt{3}+\sqrt{5}}{3\sqrt{3}-\sqrt{5}} \times \frac{3\sqrt{3}+\sqrt{5}}{3\sqrt{3}+\sqrt{5}} = \frac{(3\sqrt{3}+\sqrt{5})^2}{(3\sqrt{3})^2 - (\sqrt{5})^2}$$ 5. **Calculate the denominator:** $$ (3\sqrt{3})^2 - (\sqrt{5})^2 = 9 \times 3 - 5 = 27 - 5 = 22 $$ 6. **Expand the numerator:** $$ (3\sqrt{3}+\sqrt{5})^2 = (3\sqrt{3})^2 + 2 \times 3\sqrt{3} \times \sqrt{5} + (\sqrt{5})^2 $$ $$ = 27 + 2 \times 3 \times \sqrt{15} + 5 = 27 + 6\sqrt{15} + 5 = 32 + 6\sqrt{15} $$ 7. **Write the simplified expression:** $$ \frac{32 + 6\sqrt{15}}{22} $$ 8. **Simplify the fraction by dividing numerator and denominator by 2:** $$ \frac{\cancel{2}(16 + 3\sqrt{15})}{\cancel{2}11} = \frac{16 + 3\sqrt{15}}{11} $$ **Final answer:** $$ \frac{16 + 3\sqrt{15}}{11} $$