1. **Problem:** Simplify $\left(8a^3b^9\right)^{\frac{1}{3}}$. 2. **Formula:** When raising a power to another power, multiply the exponents: $\left(x^m\right)^n = x^{mn}$. 3. **Step 1:** Apply the exponent $\frac{1}{3}$ to each factor inside the parentheses: $$\left(8a^3b^9\right)^{\frac{1}{3}} = 8^{\frac{1}{3}} \cdot \left(a^3\right)^{\frac{1}{3}} \cdot \left(b^9\right)^{\frac{1}{3}}$$ 4. **Step 2:** Simplify each term: - $8^{\frac{1}{3}} = 2$ because $2^3 = 8$. - $\left(a^3\right)^{\frac{1}{3}} = a^{3 \cdot \frac{1}{3}} = a^1 = a$. - $\left(b^9\right)^{\frac{1}{3}} = b^{9 \cdot \frac{1}{3}} = b^3$. 5. **Final answer for 1:** $$2ab^3$$ --- 6. **Problem:** Simplify $$\left(\frac{8a^6b^{-2}}{27a^{-3}b}\right)^{-\frac{2}{3}}$$. 7. **Step 1:** Simplify inside the parentheses first: $$\frac{8a^6b^{-2}}{27a^{-3}b} = \frac{8}{27} \cdot a^{6 - (-3)} \cdot b^{-2 - 1} = \frac{8}{27} \cdot a^{9} \cdot b^{-3}$$ 8. **Step 2:** Rewrite the expression: $$\left(\frac{8}{27} a^{9} b^{-3}\right)^{-\frac{2}{3}}$$ 9. **Step 3:** Apply the exponent $-\frac{2}{3}$ to each factor: $$\left(\frac{8}{27}\right)^{-\frac{2}{3}} \cdot \left(a^{9}\right)^{-\frac{2}{3}} \cdot \left(b^{-3}\right)^{-\frac{2}{3}}$$ 10. **Step 4:** Simplify each term: - $\left(\frac{8}{27}\right)^{-\frac{2}{3}} = \left(\frac{27}{8}\right)^{\frac{2}{3}}$ (because of the negative exponent). - $\left(a^{9}\right)^{-\frac{2}{3}} = a^{9 \cdot (-\frac{2}{3})} = a^{-6}$. - $\left(b^{-3}\right)^{-\frac{2}{3}} = b^{-3 \cdot (-\frac{2}{3})} = b^{2}$. 11. **Step 5:** Simplify $\left(\frac{27}{8}\right)^{\frac{2}{3}}$: - $\left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{27^{\frac{1}{3}}}{8^{\frac{1}{3}}} = \frac{3}{2}$. - Then square it: $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$. 12. **Step 6:** Combine all: $$\frac{9}{4} \cdot a^{-6} \cdot b^{2} = \frac{9b^{2}}{4a^{6}}$$ **Final answer for 2:** $$\frac{9b^{2}}{4a^{6}}$$