1. **State the problem:** Rationalize the denominator and simplify the expression $$\sqrt[4]{\frac{7}{8}}$$. 2. **Rewrite the expression:** We have $$\sqrt[4]{\frac{7}{8}} = \frac{\sqrt[4]{7}}{\sqrt[4]{8}}$$. 3. **Simplify the denominator:** Note that $$8 = 2^3$$, so $$\sqrt[4]{8} = \sqrt[4]{2^3} = 2^{\frac{3}{4}}$$. 4. **Rationalize the denominator:** To remove the fourth root from the denominator, multiply numerator and denominator by $$2^{\frac{1}{4}}$$ because $$2^{\frac{3}{4}} \times 2^{\frac{1}{4}} = 2^{1} = 2$$. 5. **Perform the multiplication:** $$\frac{\sqrt[4]{7}}{2^{\frac{3}{4}}} \times \frac{2^{\frac{1}{4}}}{2^{\frac{1}{4}}} = \frac{\sqrt[4]{7} \times 2^{\frac{1}{4}}}{2} = \frac{\sqrt[4]{7 \times 2}}{2} = \frac{\sqrt[4]{14}}{2}$$. 6. **Final answer:** $$\boxed{\frac{\sqrt[4]{14}}{2}}$$. This is the simplified form with the denominator rationalized.