1. **State the problem:** Simplify and express with positive exponents the following expressions: a) $$\left( \frac{a^{-2}}{b^{-5}} \right)^{-3}$$ b) $$\left( \frac{32a^{2}b^{-4}}{4a^{-8}b^{-2}} \right) \left( \frac{-8a^{-2}}{-3b^{-3}} \right)$$ c) $$\left( \frac{12x^{3}y^{-1}}{-8x^{-1}y^{5}} \right)^{-2}$$ 2. **Recall exponent rules:** - $$\left( \frac{x^{m}}{y^{n}} \right)^{p} = \frac{x^{mp}}{y^{np}}$$ - $$x^{a} \cdot x^{b} = x^{a+b}$$ - $$\frac{x^{a}}{x^{b}} = x^{a-b}$$ - Negative exponents: $$x^{-m} = \frac{1}{x^{m}}$$ --- ### a) Simplify $$\left( \frac{a^{-2}}{b^{-5}} \right)^{-3}$$ 3. Apply the power of a quotient rule: $$\left( \frac{a^{-2}}{b^{-5}} \right)^{-3} = \frac{a^{-2 \times (-3)}}{b^{-5 \times (-3)}} = \frac{a^{6}}{b^{15}}$$ 4. The expression is already with positive exponents. --- ### b) Simplify $$\left( \frac{32a^{2}b^{-4}}{4a^{-8}b^{-2}} \right) \left( \frac{-8a^{-2}}{-3b^{-3}} \right)$$ 5. Simplify the first fraction: $$\frac{32a^{2}b^{-4}}{4a^{-8}b^{-2}} = \frac{32}{4} \cdot \frac{a^{2}}{a^{-8}} \cdot \frac{b^{-4}}{b^{-2}} = 8 \cdot a^{2 - (-8)} \cdot b^{-4 - (-2)} = 8a^{10}b^{-2}$$ 6. Simplify the second fraction: $$\frac{-8a^{-2}}{-3b^{-3}} = \frac{-8}{-3} \cdot a^{-2} \cdot b^{3} = \frac{8}{3} a^{-2} b^{3}$$ 7. Multiply the two results: $$8a^{10}b^{-2} \times \frac{8}{3} a^{-2} b^{3} = \frac{64}{3} a^{10 + (-2)} b^{-2 + 3} = \frac{64}{3} a^{8} b^{1} = \frac{64}{3} a^{8} b$$ --- ### c) Simplify $$\left( \frac{12x^{3}y^{-1}}{-8x^{-1}y^{5}} \right)^{-2}$$ 8. Simplify inside the parentheses first: $$\frac{12x^{3}y^{-1}}{-8x^{-1}y^{5}} = \frac{12}{-8} \cdot x^{3 - (-1)} \cdot y^{-1 - 5} = -\frac{3}{2} x^{4} y^{-6}$$ 9. Now raise to the power $$-2$$: $$\left(-\frac{3}{2} x^{4} y^{-6} \right)^{-2} = \left(-\frac{3}{2}\right)^{-2} x^{4 \times (-2)} y^{-6 \times (-2)} = \left(-\frac{3}{2}\right)^{-2} x^{-8} y^{12}$$ 10. Simplify the coefficient: $$\left(-\frac{3}{2}\right)^{-2} = \left(-\frac{2}{3}\right)^{2} = \frac{4}{9}$$ 11. So the expression is: $$\frac{4}{9} x^{-8} y^{12}$$ 12. Express with positive exponents: $$\frac{4 y^{12}}{9 x^{8}}$$ --- **Final answers:** - a) $$\frac{a^{6}}{b^{15}}$$ - b) $$\frac{64}{3} a^{8} b$$ - c) $$\frac{4 y^{12}}{9 x^{8}}$$