1. **State the problem:** Simplify the expression $$\frac{(-3u^{-3}h^{3})^{5}}{12u^{-9}h^{10}}$$ and express it with only positive exponents. 2. **Apply the power to each factor inside the parentheses:** $$(-3)^{5} (u^{-3})^{5} (h^{3})^{5} = (-3)^{5} u^{-15} h^{15}$$ 3. **Calculate the powers:** $$(-3)^{5} = -243$$ So the numerator becomes: $$-243 u^{-15} h^{15}$$ 4. **Rewrite the entire fraction:** $$\frac{-243 u^{-15} h^{15}}{12 u^{-9} h^{10}}$$ 5. **Divide coefficients and apply the quotient rule for exponents:** $$\frac{-243}{12} \times u^{-15 - (-9)} \times h^{15 - 10} = \frac{-243}{12} u^{-6} h^{5}$$ 6. **Simplify the coefficient fraction:** $$\frac{-243}{12} = \frac{-81 \times 3}{4 \times 3} = \frac{-81}{4}$$ 7. **Rewrite with positive exponents:** Since $$u^{-6} = \frac{1}{u^{6}}$$, the expression becomes: $$-\frac{81}{4} \times \frac{h^{5}}{u^{6}} = -\frac{81 h^{5}}{4 u^{6}}$$ **Final answer:** $$\boxed{-\frac{81 h^{5}}{4 u^{6}}}$$