1. **State the problem:** Rewrite the expression $$\frac{9^{-3} \cdot 5^{-6}}{5^{-10} \cdot 9^4}$$ using only positive exponents and combine powers with the same base. 2. **Recall exponent rules:** - When multiplying powers with the same base, add exponents: $$a^m \cdot a^n = a^{m+n}$$ - When dividing powers with the same base, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$ - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$ 3. **Apply the rules to the numerator:** $$9^{-3} \cdot 5^{-6}$$ (no like bases to combine here yet) 4. **Apply the rules to the denominator:** $$5^{-10} \cdot 9^4$$ (also no like bases combined yet) 5. **Rewrite the entire fraction by separating bases:** $$\frac{9^{-3}}{9^4} \cdot \frac{5^{-6}}{5^{-10}}$$ 6. **Use the division rule for exponents:** $$9^{-3 - 4} \cdot 5^{-6 - (-10)} = 9^{-7} \cdot 5^{4}$$ 7. **Rewrite negative exponent as reciprocal:** $$\frac{5^4}{9^7}$$ 8. **Final answer:** $$\boxed{\frac{5^4}{9^7}}$$