1. **State the problem:** Simplify the expression $$\frac{a^3 \cdot a^{-5} \cdot b^{-4}}{(a \cdot a^{-4} \cdot b^2)^3}$$ so that all exponents are positive. 2. **Recall exponent rules:** - When multiplying like bases, add exponents: $$a^m \cdot a^n = a^{m+n}$$ - When raising a power to a power, multiply exponents: $$(a^m)^n = a^{m \cdot n}$$ - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$ 3. **Simplify numerator:** $$a^3 \cdot a^{-5} = a^{3 + (-5)} = a^{-2}$$ So numerator becomes $$a^{-2} \cdot b^{-4}$$ 4. **Simplify denominator inside the parentheses:** $$a \cdot a^{-4} = a^{1 + (-4)} = a^{-3}$$ So inside parentheses is $$a^{-3} \cdot b^2$$ 5. **Raise denominator to the power 3:** $$(a^{-3} \cdot b^2)^3 = a^{-3 \cdot 3} \cdot b^{2 \cdot 3} = a^{-9} \cdot b^6$$ 6. **Rewrite the entire expression:** $$\frac{a^{-2} \cdot b^{-4}}{a^{-9} \cdot b^6}$$ 7. **Divide like bases by subtracting exponents:** $$a^{-2 - (-9)} = a^{-2 + 9} = a^7$$ $$b^{-4 - 6} = b^{-10}$$ 8. **Final expression with positive exponents:** $$a^7 \cdot b^{-10} = \frac{a^7}{b^{10}}$$ **Answer:** $$\boxed{\frac{a^7}{b^{10}}}$$