1. **State the problem:** Simplify the expression $$\left(\frac{12x^3y^{-1}}{-8x^{-1}y^5}\right)^{-2}$$ and express the result with positive exponents. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$. - When raising a power to another power, multiply exponents: $$(a^m)^n = a^{mn}$$. 3. **Simplify inside the parentheses first:** $$\frac{12x^3y^{-1}}{-8x^{-1}y^5} = \frac{12}{-8} \cdot \frac{x^3}{x^{-1}} \cdot \frac{y^{-1}}{y^5}$$ 4. **Simplify coefficients:** $$\frac{12}{-8} = -\frac{12}{8} = -\frac{3}{2}$$ 5. **Simplify variables using exponent rules:** $$\frac{x^3}{x^{-1}} = x^{3 - (-1)} = x^{3+1} = x^4$$ $$\frac{y^{-1}}{y^5} = y^{-1 - 5} = y^{-6}$$ 6. **Combine all:** $$-\frac{3}{2} x^4 y^{-6}$$ 7. **Now raise the entire expression to the power of -2:** $$\left(-\frac{3}{2} x^4 y^{-6}\right)^{-2}$$ 8. **Apply the power to each factor:** $$\left(-\frac{3}{2}\right)^{-2} \cdot (x^4)^{-2} \cdot (y^{-6})^{-2}$$ 9. **Simplify each term:** $$\left(-\frac{3}{2}\right)^{-2} = \left(-\frac{2}{3}\right)^2 = \frac{4}{9}$$ $$ (x^4)^{-2} = x^{4 \times (-2)} = x^{-8}$$ $$ (y^{-6})^{-2} = y^{-6 \times (-2)} = y^{12}$$ 10. **Combine all:** $$\frac{4}{9} x^{-8} y^{12}$$ 11. **Express with positive exponents only:** $$\frac{4 y^{12}}{9 x^{8}}$$ **Final answer:** $$\boxed{\frac{4 y^{12}}{9 x^{8}}}$$