1. **State the problem:** Simplify and evaluate the expression $$\frac{u^{\frac{2}{3} + 1}}{4^{\frac{5}{3}}} - \frac{4^{\frac{5}{3}}}{u^{\frac{2}{3} + 1}}$$. 2. **Rewrite exponents:** Note that $$\frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3}$$. 3. **Substitute:** The expression becomes $$\frac{u^{\frac{5}{3}}}{4^{\frac{5}{3}}} - \frac{4^{\frac{5}{3}}}{u^{\frac{5}{3}}}$$. 4. **Express as a single fraction:** $$\frac{u^{\frac{5}{3}}}{4^{\frac{5}{3}}} - \frac{4^{\frac{5}{3}}}{u^{\frac{5}{3}}} = \frac{u^{\frac{5}{3}} \cdot u^{\frac{5}{3}} - 4^{\frac{5}{3}} \cdot 4^{\frac{5}{3}}}{4^{\frac{5}{3}} \cdot u^{\frac{5}{3}}} = \frac{u^{\frac{10}{3}} - 4^{\frac{10}{3}}}{4^{\frac{5}{3}} u^{\frac{5}{3}}}$$. 5. **Factor numerator as difference of powers:** $$a^{n} - b^{n} = (a - b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$$ but here, since exponents are fractional, leave as is or evaluate numerically if values are given. 6. **Given no specific value for $u$, this is the simplified form:** $$\frac{u^{\frac{10}{3}} - 4^{\frac{10}{3}}}{4^{\frac{5}{3}} u^{\frac{5}{3}}}$$. **Final answer:** $$\frac{u^{\frac{10}{3}} - 4^{\frac{10}{3}}}{4^{\frac{5}{3}} u^{\frac{5}{3}}}$$.