1. **State the problem:** Simplify the expression $$(9m^{\frac{5}{3}})(4m^{-\frac{7}{2}})$$. 2. **Use the product rule for exponents:** When multiplying terms with the same base, add the exponents. 3. **Multiply the coefficients:** $$9 \times 4 = 36$$. 4. **Add the exponents of $m$:** $$\frac{5}{3} + \left(-\frac{7}{2}\right) = \frac{5}{3} - \frac{7}{2}$$. 5. **Find a common denominator for the exponents:** The common denominator of 3 and 2 is 6. 6. **Convert the fractions:** $$\frac{5}{3} = \frac{10}{6}, \quad \frac{7}{2} = \frac{21}{6}$$. 7. **Subtract the exponents:** $$\frac{10}{6} - \frac{21}{6} = \frac{10 - 21}{6} = -\frac{11}{6}$$. 8. **Write the simplified expression:** $$36m^{-\frac{11}{6}}$$. 9. **Optional: Express with positive exponent:** $$36 \times \frac{1}{m^{\frac{11}{6}}} = \frac{36}{m^{\frac{11}{6}}}$$. **Final answer:** $$\boxed{\frac{36}{m^{\frac{11}{6}}}}$$