1. **State the problem:** Simplify the expression $$\frac{\sqrt[4]{5}}{3\sqrt[4]{4}}$$. 2. **Recall the properties of roots:** The 4th root of a number $a$ is $a^{\frac{1}{4}}$. 3. **Rewrite the expression using exponents:** $$\frac{5^{\frac{1}{4}}}{3 \times 4^{\frac{1}{4}}}$$ 4. **Combine the roots in numerator and denominator:** $$\frac{5^{\frac{1}{4}}}{3 \times 4^{\frac{1}{4}}} = \frac{5^{\frac{1}{4}}}{3 \times (2^2)^{\frac{1}{4}}} = \frac{5^{\frac{1}{4}}}{3 \times 2^{\frac{2}{4}}} = \frac{5^{\frac{1}{4}}}{3 \times 2^{\frac{1}{2}}}$$ 5. **Express the denominator's root as a square root:** $$2^{\frac{1}{2}} = \sqrt{2}$$ 6. **Rewrite the expression:** $$\frac{5^{\frac{1}{4}}}{3 \sqrt{2}}$$ 7. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{2}$: $$\frac{5^{\frac{1}{4}}}{3 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5^{\frac{1}{4}} \sqrt{2}}{3 \times 2} = \frac{5^{\frac{1}{4}} \sqrt{2}}{6}$$ 8. **Final simplified form:** $$\boxed{\frac{\sqrt{2} \sqrt[4]{5}}{6}}$$