1. **Problem:** Simplify the expression $$ (15^{\frac{3}{8}})(81^{\frac{1}{34}})(15^{\frac{3}{4}})(81^{\frac{1}{7}})(15^{\frac{5}{8}})(81^{\frac{1}{7}}) $$ 2. **Formula and rules:** When multiplying powers with the same base, add the exponents: $$ a^m \times a^n = a^{m+n} $$ 3. **Step-by-step simplification:** - Group terms with the same base: $$ (15^{\frac{3}{8}} \times 15^{\frac{3}{4}} \times 15^{\frac{5}{8}}) \times (81^{\frac{1}{34}} \times 81^{\frac{1}{7}} \times 81^{\frac{1}{7}}) $$ - Add exponents for base 15: $$ \frac{3}{8} + \frac{3}{4} + \frac{5}{8} = \frac{3}{8} + \frac{6}{8} + \frac{5}{8} = \frac{14}{8} = \frac{7}{4} $$ - Add exponents for base 81: $$ \frac{1}{34} + \frac{1}{7} + \frac{1}{7} = \frac{1}{34} + \frac{2}{7} = \frac{1}{34} + \frac{68}{238} = \frac{7}{238} + \frac{68}{238} = \frac{75}{238} $$ - So the expression becomes: $$ 15^{\frac{7}{4}} \times 81^{\frac{75}{238}} $$ 4. **Final simplified form:** $$ \boxed{15^{\frac{7}{4}} \times 81^{\frac{75}{238}}} $$