Exponent Simplification A5C2Dd

1. **State the problem:** Simplify the expression $$\left(\frac{a^{-2} b^{4} c^{5}}{a^{3} b^{-2} d^{3}}\right)^{4} \div \left(\frac{a^{-5} c^{3} d^{-1}}{a^{3} b^{2} c^{3}}\right)^{-5}$$ and verify the final simplified form. 2. **Recall exponent rules:** - When dividing powers with the same base, subtract exponents: $$a^{m} \div a^{n} = a^{m-n}$$ - When raising a power to another power, multiply exponents: $$(a^{m})^{n} = a^{mn}$$ - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^{m}}$$ - When dividing by a fraction, multiply by its reciprocal. 3. **Simplify inside the first parentheses:** $$\frac{a^{-2} b^{4} c^{5}}{a^{3} b^{-2} d^{3}} = a^{-2-3} b^{4-(-2)} c^{5} d^{-3} = a^{-5} b^{6} c^{5} d^{-3}$$ 4. **Raise the result to the 4th power:** $$\left(a^{-5} b^{6} c^{5} d^{-3}\right)^{4} = a^{-20} b^{24} c^{20} d^{-12}$$ 5. **Simplify inside the second parentheses:** $$\frac{a^{-5} c^{3} d^{-1}}{a^{3} b^{2} c^{3}} = a^{-5-3} b^{-2} c^{3-3} d^{-1} = a^{-8} b^{-2} d^{-1}$$ 6. **Raise this to the power of -5:** $$\left(a^{-8} b^{-2} d^{-1}\right)^{-5} = a^{40} b^{10} d^{5}$$ 7. **Divide the first result by the second:** $$a^{-20} b^{24} c^{20} d^{-12} \div a^{40} b^{10} d^{5} = a^{-20-40} b^{24-10} c^{20} d^{-12-5} = a^{-60} b^{14} c^{20} d^{-17}$$ 8. **Rewrite negative exponents as positive in denominator:** $$\frac{b^{14} c^{20}}{a^{60} d^{17}}$$ **Final answer:** $$\boxed{\frac{b^{14} c^{20}}{a^{60} d^{17}}}$$ This matches the user's final answer.