1. **State the problem:** Simplify the expression $$\frac{15 e^{-2} f^{3}}{27 e^{-3} f^{-6}}$$. 2. **Recall the rules for exponents:** - When dividing like bases, subtract the exponents: $$a^{m} \div a^{n} = a^{m-n}$$. - When multiplying like bases, add the exponents: $$a^{m} \times a^{n} = a^{m+n}$$. 3. **Simplify the coefficients:** $$\frac{15}{27} = \frac{15 \div 3}{27 \div 3} = \frac{5}{9}$$ 4. **Simplify the $e$ terms:** $$e^{-2} \div e^{-3} = e^{-2 - (-3)} = e^{-2 + 3} = e^{1}$$ 5. **Simplify the $f$ terms:** $$f^{3} \div f^{-6} = f^{3 - (-6)} = f^{3 + 6} = f^{9}$$ 6. **Combine all simplified parts:** $$\frac{15 e^{-2} f^{3}}{27 e^{-3} f^{-6}} = \frac{5}{9} \times e^{1} \times f^{9} = \frac{5}{9} e f^{9}$$ **Final answer:** $$\frac{5}{9} e f^{9}$$