1. **Problem (a):** Find the value of $$\left(\frac{27}{8}\right)^{-\frac{1}{3}} \left(\frac{81}{16}\right)^{\frac{1}{4}} \div \left(\frac{4}{25}\right)^{-\frac{1}{2}}$$ 2. **Step 1:** Express each term with prime bases. - $$\frac{27}{8} = \frac{3^3}{2^3}$$ - $$\frac{81}{16} = \frac{3^4}{2^4}$$ - $$\frac{4}{25} = \frac{2^2}{5^2}$$ 3. **Step 2:** Apply the exponents: - $$\left(\frac{27}{8}\right)^{-\frac{1}{3}} = \left(\frac{3^3}{2^3}\right)^{-\frac{1}{3}} = \frac{3^{-1}}{2^{-1}} = \frac{2}{3}$$ - $$\left(\frac{81}{16}\right)^{\frac{1}{4}} = \left(\frac{3^4}{2^4}\right)^{\frac{1}{4}} = \frac{3}{2}$$ - $$\left(\frac{4}{25}\right)^{-\frac{1}{2}} = \left(\frac{2^2}{5^2}\right)^{-\frac{1}{2}} = \frac{5}{2}$$ 4. **Step 3:** Substitute back and simplify: $$\frac{2}{3} \times \frac{3}{2} \div \frac{5}{2} = 1 \times \frac{2}{5} = \frac{2}{5}$$ **Answer (a):** $$\frac{2}{5}$$ --- 1. **Problem (b):** Find the value of $$\left(\frac{25}{16}\right)^{-\frac{1}{3}} \left(\frac{125}{64}\right)^{\frac{1}{3}} \div \left(\frac{8}{27}\right)^{-\frac{1}{3}}$$ 2. **Step 1:** Express bases: - $$\frac{25}{16} = \frac{5^2}{2^4}$$ - $$\frac{125}{64} = \frac{5^3}{2^6}$$ - $$\frac{8}{27} = \frac{2^3}{3^3}$$ 3. **Step 2:** Apply exponents: - $$\left(\frac{25}{16}\right)^{-\frac{1}{3}} = \left(\frac{5^2}{2^4}\right)^{-\frac{1}{3}} = \frac{5^{-\frac{2}{3}}}{2^{-\frac{4}{3}}} = \frac{2^{\frac{4}{3}}}{5^{\frac{2}{3}}}$$ - $$\left(\frac{125}{64}\right)^{\frac{1}{3}} = \frac{5}{2^2} = \frac{5}{4}$$ - $$\left(\frac{8}{27}\right)^{-\frac{1}{3}} = \left(\frac{2^3}{3^3}\right)^{-\frac{1}{3}} = \frac{3}{2}$$ 4. **Step 3:** Substitute and simplify: $$\frac{2^{\frac{4}{3}}}{5^{\frac{2}{3}}} \times \frac{5}{4} \div \frac{3}{2} = \frac{2^{\frac{4}{3}}}{5^{\frac{2}{3}}} \times \frac{5}{4} \times \frac{2}{3} = \frac{2^{\frac{4}{3}+1}}{4 \times 3} \times \frac{5^{1 - \frac{2}{3}}}{1} = \frac{2^{\frac{7}{3}}}{12} \times 5^{\frac{1}{3}}$$ 5. **Step 4:** Simplify powers: - $$2^{\frac{7}{3}} = 2^{2 + \frac{1}{3}} = 4 \times 2^{\frac{1}{3}}$$ 6. **Step 5:** Final expression: $$\frac{4 \times 2^{\frac{1}{3}}}{12} \times 5^{\frac{1}{3}} = \frac{1}{3} \times (2^{\frac{1}{3}} \times 5^{\frac{1}{3}}) = \frac{1}{3} (10^{\frac{1}{3}})$$ **Answer (b):** $$\frac{\sqrt[3]{10}}{3}$$ --- 1. **Problem (c):** Find the value of $$\left(\frac{27}{8}\right)^{\frac{1}{3}} \left[ \left(\frac{243}{32}\right)^{\frac{1}{5}} \div \left(\frac{2}{3}\right)^{-2} \right]$$ 2. **Step 1:** Express bases: - $$\frac{27}{8} = \frac{3^3}{2^3}$$ - $$\frac{243}{32} = \frac{3^5}{2^5}$$ - $$\frac{2}{3}$$ 3. **Step 2:** Apply exponents: - $$\left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{3}{2}$$ - $$\left(\frac{243}{32}\right)^{\frac{1}{5}} = \frac{3}{2}$$ - $$\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ 4. **Step 3:** Evaluate the bracket: $$\frac{3}{2} \div \frac{9}{4} = \frac{3}{2} \times \frac{4}{9} = \frac{2}{3}$$ 5. **Step 4:** Multiply outside term: $$\frac{3}{2} \times \frac{2}{3} = 1$$ **Answer (c):** $$1$$ --- 1. **Problem (d):** Find the value of $$\frac{3^4 \times 27^3 \times 9^5}{81^6 \times 3^3 \times 9^{-2}}$$ 2. **Step 1:** Express all terms with base 3: - $$27 = 3^3$$ - $$9 = 3^2$$ - $$81 = 3^4$$ 3. **Step 2:** Rewrite expression: $$\frac{3^4 \times (3^3)^3 \times (3^2)^5}{(3^4)^6 \times 3^3 \times (3^2)^{-2}} = \frac{3^4 \times 3^{9} \times 3^{10}}{3^{24} \times 3^3 \times 3^{-4}}$$ 4. **Step 3:** Simplify exponents: - Numerator: $$3^{4+9+10} = 3^{23}$$ - Denominator: $$3^{24+3-4} = 3^{23}$$ 5. **Step 4:** Final simplification: $$\frac{3^{23}}{3^{23}} = 1$$ **Answer (d):** $$1$$