1. Problem: Simplify $\frac{x^{-4} \cdot x^{3}}{x}$. 2. Use the laws of exponents: $a^{m} \cdot a^{n} = a^{m+n}$ and $\frac{a^{m}}{a^{n}} = a^{m-n}$. 3. Simplify numerator: $x^{-4} \cdot x^{3} = x^{-4+3} = x^{-1}$. 4. Now divide by $x = x^{1}$: $$\frac{x^{-1}}{x^{1}} = x^{-1-1} = x^{-2} = \frac{1}{x^{2}}.$$ This matches choice C. --- 1. Problem: Simplify $(x^{2})^{-1} \cdot x^{2}$. 2. Use the power of a power rule: $(a^{m})^{n} = a^{mn}$. 3. Simplify: $(x^{2})^{-1} = x^{2 \times (-1)} = x^{-2}$. 4. Multiply: $x^{-2} \cdot x^{2} = x^{-2+2} = x^{0} = 1.$ This matches choice I. --- 1. Problem: Simplify $\left(\frac{x^{-5}}{x^{-2}}\right)^{-1}$. 2. Use quotient rule: $\frac{a^{m}}{a^{n}} = a^{m-n}$. 3. Simplify inside parentheses: $$\frac{x^{-5}}{x^{-2}} = x^{-5 - (-2)} = x^{-5 + 2} = x^{-3}.$$ 4. Now raise to $-1$: $(x^{-3})^{-1} = x^{(-3) \times (-1)} = x^{3}.$ This matches choice A. --- 1. Problem: Simplify $\frac{(x^{-3})^{2}}{(x^{2})^{-1}}$. 2. Use power of power rule: $(a^{m})^{n} = a^{mn}$. 3. Simplify numerator: $(x^{-3})^{2} = x^{-6}$. 4. Simplify denominator: $(x^{2})^{-1} = x^{-2}$. 5. Divide: $$\frac{x^{-6}}{x^{-2}} = x^{-6 - (-2)} = x^{-6 + 2} = x^{-4} = \frac{1}{x^{4}}.$$ This matches choice E. --- Final code: C I A E