1. **Problem statement:** Simplify each expression given in parts 8 and 9. --- ### Part 8 2. **Expression a:** Simplify $8a\ (3xy^2)^3$. 3. Use the power of a product rule: $ (abc)^n = a^n b^n c^n $. 4. Calculate: $$ (3xy^2)^3 = 3^3 \times x^3 \times (y^2)^3 = 27x^3y^6 $$ 5. Multiply by $8a$: $$ 8a \times 27x^3y^6 = 216ax^3y^6 $$ --- 6. **Expression b:** Simplify $8a\ (2a^2b^3)^5$. 7. Apply power to each factor: $$ (2a^2b^3)^5 = 2^5 \times (a^2)^5 \times (b^3)^5 = 32a^{10}b^{15} $$ 8. Multiply by $8a$: $$ 8a \times 32a^{10}b^{15} = 256a^{11}b^{15} $$ --- 9. **Expression c:** Simplify $8a\ (5x^4y^3)^2$. 10. Apply power: $$ (5x^4y^3)^2 = 5^2 \times (x^4)^2 \times (y^3)^2 = 25x^8y^6 $$ 11. Multiply by $8a$: $$ 8a \times 25x^8y^6 = 200ax^8y^6 $$ --- 12. **Expression d:** Simplify $8a\ (10x^5y^7)^3$. 13. Apply power: $$ (10x^5y^7)^3 = 10^3 \times (x^5)^3 \times (y^7)^3 = 1000x^{15}y^{21} $$ 14. Multiply by $8a$: $$ 8a \times 1000x^{15}y^{21} = 8000ax^{15}y^{21} $$ --- ### Part 9 15. **Expression a:** Simplify $\frac{3x^2 + 5x^2}{2x^2}$. 16. Combine numerator: $$ 3x^2 + 5x^2 = 8x^2 $$ 17. So expression is: $$ \frac{8x^2}{2x^2} $$ 18. Cancel $x^2$: $$ \frac{\cancel{8} \cancel{x^2}}{\cancel{2} \cancel{x^2}} = \frac{8}{2} = 4 $$ --- 19. **Expression b:** Simplify $\frac{15x^{10}}{3x^4} + 8x^6$. 20. Simplify fraction: $$ \frac{15x^{10}}{3x^4} = \frac{15}{3} x^{10-4} = 5x^6 $$ 21. Add $8x^6$: $$ 5x^6 + 8x^6 = 13x^6 $$ --- 22. **Expression c:** Simplify $\frac{(2x^2)^3 \times (3x)^2}{2x^5}$. 23. Calculate powers: $$ (2x^2)^3 = 2^3 x^{6} = 8x^6 $$ $$ (3x)^2 = 3^2 x^2 = 9x^2 $$ 24. Multiply numerator: $$ 8x^6 \times 9x^2 = 72x^{8} $$ 25. Expression becomes: $$ \frac{72x^8}{2x^5} $$ 26. Simplify fraction: $$ \frac{72}{2} x^{8-5} = 36x^3 $$ --- 27. **Expression d:** Simplify $\frac{5a^2b^4}{3c} \times \frac{abc}{15a^2b^2}$. 28. Multiply numerators and denominators: $$ \frac{5a^2b^4 \times abc}{3c \times 15a^2b^2} = \frac{5a^2b^4 \times a b c}{45 a^2 b^2 c} $$ 29. Simplify numerator: $$ 5a^2b^4 \times a b c = 5 a^{3} b^{5} c $$ 30. Expression: $$ \frac{5 a^{3} b^{5} c}{45 a^{2} b^{2} c} $$ 31. Cancel common factors: $$ \frac{\cancel{5} a^{3-2} b^{5-2} \cancel{c}}{\cancel{45} \cancel{a^{2}} \cancel{b^{2}} \cancel{c}} = \frac{1 a^{1} b^{3}}{9} = \frac{a b^{3}}{9} $$ --- 32. **Expression e:** Simplify $\frac{4a^5b}{cd} \div \frac{a}{c^2 d^3}$. 33. Division of fractions: $$ \frac{4a^5b}{cd} \times \frac{c^2 d^3}{a} $$ 34. Multiply numerators and denominators: $$ \frac{4a^5 b c^2 d^3}{c d a} $$ 35. Simplify denominator: $$ c d a $$ 36. Cancel common factors: $$ \frac{4 \cancel{a^5} b c^2 d^3}{\cancel{c} \cancel{d} \cancel{a}} = 4 a^{5-1} b c^{2-1} d^{3-1} = 4 a^{4} b c d^{2} $$ --- 37. **Expression f:** Simplify $\frac{(4x^2 y^3)^3}{2x^5 y^2}$. 38. Calculate numerator: $$ (4x^2 y^3)^3 = 4^3 x^{6} y^{9} = 64 x^{6} y^{9} $$ 39. Expression: $$ \frac{64 x^{6} y^{9}}{2 x^{5} y^{2}} $$ 40. Simplify fraction: $$ \frac{64}{2} x^{6-5} y^{9-2} = 32 x^{1} y^{7} = 32 x y^{7} $$ --- 41. **Expression g:** Simplify $\frac{(2x^2 y)^5}{(2 x y)^3}$. 42. Calculate numerator: $$ (2 x^{2} y)^5 = 2^{5} x^{10} y^{5} = 32 x^{10} y^{5} $$ 43. Calculate denominator: $$ (2 x y)^3 = 2^{3} x^{3} y^{3} = 8 x^{3} y^{3} $$ 44. Expression: $$ \frac{32 x^{10} y^{5}}{8 x^{3} y^{3}} $$ 45. Simplify fraction: $$ \frac{32}{8} x^{10-3} y^{5-3} = 4 x^{7} y^{2} $$ --- 46. **Expression h:** Simplify $(((2 x y^{2})^{2})^{2})^{2}$. 47. Simplify stepwise: $$ (2 x y^{2})^{2} = 2^{2} x^{2} y^{4} = 4 x^{2} y^{4} $$ 48. Raise to power 2 again: $$ (4 x^{2} y^{4})^{2} = 4^{2} x^{4} y^{8} = 16 x^{4} y^{8} $$ 49. Raise to power 2 once more: $$ (16 x^{4} y^{8})^{2} = 16^{2} x^{8} y^{16} = 256 x^{8} y^{16} $$ --- **Final answers:** - 8a: $216 a x^{3} y^{6}$ - 8b: $256 a^{11} b^{15}$ - 8c: $200 a x^{8} y^{6}$ - 8d: $8000 a x^{15} y^{21}$ - 9a: $4$ - 9b: $13 x^{6}$ - 9c: $36 x^{3}$ - 9d: $\frac{a b^{3}}{9}$ - 9e: $4 a^{4} b c d^{2}$ - 9f: $32 x y^{7}$ - 9g: $4 x^{7} y^{2}$ - 9h: $256 x^{8} y^{16}$