Exponent Fraction Simplify 54A1C4

1. **Problem Statement:** Simplify the expression $$\left(\frac{a^3 b^{-1} c}{a^4 b^{-3} d^2}\right)^4 \div \left(\frac{a^{-3} b^4 c^2 d^{-1}}{b^3 c^{-2} d^{-3}}\right)^{-4}$$ and verify the result. 2. **Formula and Rules:** - When raising a fraction to a power, raise numerator and denominator to that power. - Division of expressions is multiplication by the reciprocal. - Use the property $$\frac{x^m}{x^n} = x^{m-n}$$ for same bases. - Negative exponents mean reciprocal: $$x^{-m} = \frac{1}{x^m}$$. 3. **Step-by-step Simplification:** - Simplify inside the first parentheses: $$\frac{a^3 b^{-1} c}{a^4 b^{-3} d^2} = a^{3-4} b^{-1 - (-3)} c^{1} d^{-2} = a^{-1} b^{2} c^{1} d^{-2}$$ - Raise to the 4th power: $$\left(a^{-1} b^{2} c^{1} d^{-2}\right)^4 = a^{-4} b^{8} c^{4} d^{-8}$$ - Simplify inside the second parentheses: $$\frac{a^{-3} b^{4} c^{2} d^{-1}}{b^{3} c^{-2} d^{-3}} = a^{-3} b^{4-3} c^{2 - (-2)} d^{-1 - (-3)} = a^{-3} b^{1} c^{4} d^{2}$$ - Raise to the power $$-4$$: $$\left(a^{-3} b^{1} c^{4} d^{2}\right)^{-4} = a^{12} b^{-4} c^{-16} d^{-8}$$ - Now divide the first result by the second (equivalent to multiply by reciprocal): $$a^{-4} b^{8} c^{4} d^{-8} \times a^{-12} b^{4} c^{16} d^{8} = a^{-16} b^{12} c^{20} d^{0}$$ - Simplify: $$a^{-16} b^{12} c^{20}$$ - Express with positive exponents in denominator: $$\frac{b^{12} c^{20}}{a^{16}}$$ - **Final answer:** $$\frac{b^{12} c^{20}}{a^{16}}$$ --- 4. **Second Problem:** Simplify $$\left(\frac{a^{5} b^{-3} c^{2}}{a^{-1} b^{2} d^{3}}\right)^{-3} \div \left(\frac{a^{3} b^{-2} d^{3}}{b^{3} c^{-2} d^{-1}}\right)^{5}$$ - Simplify inside first parentheses: $$\frac{a^{5} b^{-3} c^{2}}{a^{-1} b^{2} d^{3}} = a^{5 - (-1)} b^{-3 - 2} c^{2} d^{-3} = a^{6} b^{-5} c^{2} d^{-3}$$ - Raise to $$-3$$ power: $$\left(a^{6} b^{-5} c^{2} d^{-3}\right)^{-3} = a^{-18} b^{15} c^{-6} d^{9}$$ - Simplify inside second parentheses: $$\frac{a^{3} b^{-2} d^{3}}{b^{3} c^{-2} d^{-1}} = a^{3} b^{-2 - 3} c^{2} d^{3 - (-1)} = a^{3} b^{-5} c^{2} d^{4}$$ - Raise to 5th power: $$\left(a^{3} b^{-5} c^{2} d^{4}\right)^{5} = a^{15} b^{-25} c^{10} d^{20}$$ - Divide first by second (multiply by reciprocal): $$a^{-18} b^{15} c^{-6} d^{9} \times a^{-15} b^{25} c^{-10} d^{-20} = a^{-33} b^{40} c^{-16} d^{-11}$$ - Express with positive exponents in denominator: $$\frac{b^{40}}{a^{33} c^{16} d^{11}}$$ - **Final answer:** $$\frac{b^{40}}{a^{33} c^{16} d^{11}}$$ --- 5. **Third Problem:** Simplify $$\left(\frac{a^{-2} b^{3} c^{2} d^{-1}}{a^{3} b^{4} c^{-1} d^{3}}\right)^3 \div \left(\frac{a^{3} b^{-2} d^{2}}{a^{5} c^{3} d^{-3}}\right)^{-2}$$ - Simplify inside first parentheses: $$\frac{a^{-2} b^{3} c^{2} d^{-1}}{a^{3} b^{4} c^{-1} d^{3}} = a^{-2 - 3} b^{3 - 4} c^{2 - (-1)} d^{-1 - 3} = a^{-5} b^{-1} c^{3} d^{-4}$$ - Raise to 3rd power: $$\left(a^{-5} b^{-1} c^{3} d^{-4}\right)^3 = a^{-15} b^{-3} c^{9} d^{-12}$$ - Simplify inside second parentheses: $$\frac{a^{3} b^{-2} d^{2}}{a^{5} c^{3} d^{-3}} = a^{3 - 5} b^{-2} c^{-3} d^{2 - (-3)} = a^{-2} b^{-2} c^{-3} d^{5}$$ - Raise to $$-2$$ power: $$\left(a^{-2} b^{-2} c^{-3} d^{5}\right)^{-2} = a^{4} b^{4} c^{6} d^{-10}$$ - Divide first by second (multiply by reciprocal): $$a^{-15} b^{-3} c^{9} d^{-12} \times a^{-4} b^{-4} c^{-6} d^{10} = a^{-19} b^{-7} c^{3} d^{-2}$$ - Express with positive exponents in denominator: $$\frac{c^{3}}{a^{19} b^{7} d^{2}}$$ - **Final answer:** $$\frac{c^{3}}{a^{19} b^{7} d^{2}}$$ --- 6. **Summary:** - First expression simplifies to $$\frac{b^{12} c^{20}}{a^{16}}$$ - Second expression simplifies to $$\frac{b^{40}}{a^{33} c^{16} d^{11}}$$ - Third expression simplifies to $$\frac{c^{3}}{a^{19} b^{7} d^{2}}$$ These results are consistent with the method used in the user's original calculations, confirming correctness.