1. The problem is to simplify the expression $$(243x^4)^{\frac{4}{5}}$$. 2. We use the power of a product rule: $$(ab)^m = a^m b^m$$. 3. Apply the rule: $$ (243)^{\frac{4}{5}} (x^4)^{\frac{4}{5}} $$. 4. Simplify each part separately. 5. For the number: $$243 = 3^5$$, so $$ (243)^{\frac{4}{5}} = (3^5)^{\frac{4}{5}} = 3^{5 \times \frac{4}{5}} = 3^4 = 81$$. 6. For the variable: $$(x^4)^{\frac{4}{5}} = x^{4 \times \frac{4}{5}} = x^{\frac{16}{5}}$$. 7. Combine the results: $$81 x^{\frac{16}{5}}$$. 8. Express $x^{\frac{16}{5}}$ as $x^{3 + \frac{1}{5}} = x^3 x^{\frac{1}{5}} = x^3 \sqrt[5]{x}$$. 9. Final simplified expression: $$81 x^3 \sqrt[5]{x}$$.