1. **State the problem:** Rationalize the denominator of the expression $$\frac{ \sqrt[3]{27} \times \sqrt[4]{16} }{ \sqrt[5]{16} \times \sqrt{9} }.$$\n\n2. **Simplify each root:**\n- $$\sqrt[3]{27} = 3$$ because $27 = 3^3$.\n- $$\sqrt[4]{16} = 2$$ because $16 = 2^4$.\n- $$\sqrt[5]{16} = \sqrt[5]{2^4} = 2^{\frac{4}{5}}$$ (leave as is for now).\n- $$\sqrt{9} = 3$$ because $9 = 3^2$.\n\n3. **Rewrite the expression with simplified terms:**\n$$\frac{3 \times 2}{2^{\frac{4}{5}} \times 3} = \frac{6}{3 \times 2^{\frac{4}{5}}}.$$\n\n4. **Cancel common factors:**\nThe numerator and denominator both have a factor of 3, so cancel 3:\n$$\frac{6}{3 \times 2^{\frac{4}{5}}} = \frac{2 \times 3}{3 \times 2^{\frac{4}{5}}} = \frac{2}{2^{\frac{4}{5}}}.$$\n\n5. **Simplify the fraction involving powers of 2:**\n$$\frac{2}{2^{\frac{4}{5}}} = 2^{1 - \frac{4}{5}} = 2^{\frac{1}{5}} = \sqrt[5]{2}.$$\n\n6. **Final answer:**\n$$\boxed{\sqrt[5]{2}}.$$\n\n**Matching the multiple choice:** The correct choice for the simplified expression is option A: $$\frac{3(4)}{\sqrt[5]{16} \times \sqrt{9}}$$ (since $3 \times 4 = 12$ is a misprint, the correct simplification is $3 \times 2$ which matches option B, but the final simplified answer matches option A's final answer $$\sqrt[5]{2}$$).\n\nTherefore, the final answer is option A: $$\sqrt[5]{2}$$.