1. Problem (a): Simplify $32x^8 \div 8x^{32}$. 2. Use the division rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$. 3. Simplify the coefficients: $\frac{32}{8} = \cancel{\frac{32}{8}}4$. 4. Simplify the variables: $x^{8-32} = x^{-24}$. 5. Combine results: $4x^{-24} = \frac{4}{x^{24}}$. 6. Problem (b): Simplify $\left( \frac{x^3}{64} \right)^{\frac{2}{3}}$. 7. Use the power of a quotient rule: $\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$. 8. Apply the exponent: $\frac{(x^3)^{\frac{2}{3}}}{64^{\frac{2}{3}}}$. 9. Simplify numerator: $(x^3)^{\frac{2}{3}} = x^{3 \times \frac{2}{3}} = x^2$. 10. Simplify denominator: $64^{\frac{2}{3}} = (4^3)^{\frac{2}{3}} = 4^{3 \times \frac{2}{3}} = 4^2 = 16$. 11. Final answer: $\frac{x^2}{16}$. 12. Problem (c): Simplify $(3x^3)^3$. 13. Use the power of a product rule: $(ab)^m = a^m b^m$. 14. Apply exponent: $3^3 \times (x^3)^3 = 27 \times x^{3 \times 3} = 27x^9$. 15. Final answer: $27x^9$.