1. **Problem statement:** Simplify each expression using exponent laws and express answers with positive exponents. 2. **Recall exponent laws:** - Product of powers: $a^m \cdot a^n = a^{m+n}$ - Power of a power: $(a^m)^n = a^{m \cdot n}$ - Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$ - Negative exponent: $a^{-m} = \frac{1}{a^m}$ --- **a) Simplify** $(x^3)(x^{-\frac{2}{3}})$ 3. Apply product of powers: $$x^{3 + (-\frac{2}{3})} = x^{3 - \frac{2}{3}}$$ 4. Convert 3 to fraction: $3 = \frac{9}{3}$ 5. Subtract exponents: $$x^{\frac{9}{3} - \frac{2}{3}} = x^{\frac{7}{3}}$$ 6. The exponent is positive, so final answer: $$x^{\frac{7}{3}}$$ --- **b) Simplify** $(81^{-0.25})^3$ 7. Use power of a power: $$81^{-0.25 \cdot 3} = 81^{-0.75}$$ 8. Express 81 as power of 3: $81 = 3^4$ 9. Substitute: $$(3^4)^{-0.75} = 3^{4 \times (-0.75)} = 3^{-3}$$ 10. Convert negative exponent: $$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$$ --- **c) Simplify** $\frac{(m^{-2})^{\frac{2}{3}}}{(m^{\frac{1}{2}})^4}$ 11. Apply power of a power to numerator: $$(m^{-2})^{\frac{2}{3}} = m^{-2 \times \frac{2}{3}} = m^{-\frac{4}{3}}$$ 12. Apply power of a power to denominator: $$(m^{\frac{1}{2}})^4 = m^{\frac{1}{2} \times 4} = m^2$$ 13. Use quotient of powers: $$\frac{m^{-\frac{4}{3}}}{m^2} = m^{-\frac{4}{3} - 2}$$ 14. Convert 2 to fraction: $2 = \frac{6}{3}$ 15. Subtract exponents: $$m^{-\frac{4}{3} - \frac{6}{3}} = m^{-\frac{10}{3}}$$ 16. Convert negative exponent: $$m^{-\frac{10}{3}} = \frac{1}{m^{\frac{10}{3}}}$$ --- **Final answers:** - a) $x^{\frac{7}{3}}$ - b) $\frac{1}{27}$ - c) $\frac{1}{m^{\frac{10}{3}}}$