1. Problem: Simplify $$\frac{(5x - 6y)^5}{(10x - 12y)^{-3}}$$. 2. Note that $$10x - 12y = 2(5x - 6y)$$. 3. Rewrite denominator using the factor: $$(10x - 12y)^{-3} = \left(2(5x - 6y)\right)^{-3} = 2^{-3}(5x - 6y)^{-3}$$. 4. So the original expression becomes: $$\frac{(5x - 6y)^5}{2^{-3}(5x - 6y)^{-3}} = (5x - 6y)^5 \times 2^3 (5x - 6y)^3 = 2^3 (5x - 6y)^{5+3} = 8 (5x - 6y)^8$$. 5. The correct choice is (a) $8(5x - 6y)^8$. 6. Problem: Simplify $$\frac{(4x - 6y)^{-2}}{(6x - 9y)^{-4}}$$. 7. Factor inside terms: $$4x - 6y = 2(2x - 3y), \quad 6x - 9y = 3(2x - 3y)$$. 8. Rewrite expression: $$\frac{[2(2x - 3y)]^{-2}}{[3(2x - 3y)]^{-4}} = \frac{2^{-2}(2x - 3y)^{-2}}{3^{-4}(2x - 3y)^{-4}} = 2^{-2} (2x - 3y)^{-2} \times 3^{4} (2x - 3y)^{4} = 3^{4} 2^{-2} (2x - 3y)^{2}$$. 9. Calculate numeric coefficients: $$3^{4} = 81, \quad 2^{-2} = \frac{1}{4}$$. 10. So the expression simplifies to: $$\frac{81}{4} (2x - 3y)^2$$. 11. Correct choice is (c) $\frac{81}{4} (2x - 3y)^2$. 12. Problem: Multiply $(2x^2 + 3xy - y^3)(x - 5y)$. 13. Distribute each term: - $2x^2 \times x = 2x^3$ - $2x^2 \times (-5y) = -10x^2y$ - $3xy \times x = 3x^2y$ - $3xy \times (-5y) = -15xy^2$ - $(-y^3) \times x = -xy^3$ - $(-y^3) \times (-5y) = 5y^4$ 14. Combine like terms: $$2x^3 + (-10x^2y + 3x^2y) - 15xy^2 - xy^3 + 5y^4 = 2x^3 - 7x^2y - 15xy^2 - xy^3 + 5y^4$$. 15. The correct choice is (b). 16. Problem: Multiply $(3a - 2b)(9a^2 + 6ab + 4b^2)$. 17. Expand: - $3a \times 9a^2 = 27a^3$ - $3a \times 6ab = 18a^2b$ - $3a \times 4b^2 = 12ab^2$ - $-2b \times 9a^2 = -18a^2b$ - $-2b \times 6ab = -12ab^2$ - $-2b \times 4b^2 = -8b^3$ 18. Combine like terms: $27a^3 + (18a^2b - 18a^2b) + (12ab^2 - 12ab^2) - 8b^3 = 27a^3 - 8b^3$ 19. Correct choice is (b). 20. Problem: Simplify $(u + 4)(u^2 - 4uv + 16v^2)$. 21. Recognize $u^3 + 64v^3 = (u + 4)(u^2 - 4uv + 16v^2)$ as sum of cubes factorization. 22. Therefore, $(u + 4)(u^2 - 4uv + 16v^2) = u^3 + 64v^3$. 23. Correct choice is (a). 24. Problem: Simplify $3y^2z^3 (2y - 3z)(2y + 3z)$. 25. Recognize difference of squares: $$(2y - 3z)(2y + 3z) = (2y)^2 - (3z)^2 = 4y^2 - 9z^2$$. 26. Multiply: $$3y^2z^3 (4y^2 - 9z^2) = 3y^2z^3 \times 4y^2 - 3y^2z^3 \times 9z^2 = 12 y^{4} z^{3} - 27 y^{2} z^{5}$$. 27. Correct choice is (d) none, since no option matches $12y^4z^3 - 27y^2z^5$. 28. Problem: Simplify $2xy^5 (3x + y)(3x - y)$. 29. Difference of squares: $$(3x + y)(3x - y) = (3x)^2 - y^2 = 9x^2 - y^2$$. 30. Multiply: $$2xy^5 (9x^2 - y^2) = 18 x^{3} y^{5} - 2 x y^{7}$$. 31. Correct choice is (b). 32. Problem: Simplify $(x^2 - 4x + 4)(x^2 + 4x + 4)$. 33. Recognize $(x^2 - 4x + 4) = (x - 2)^2$ and $(x^2 + 4x + 4) = (x + 2)^2$. 34. So product is: $$ (x - 2)^2 (x + 2)^2 = ig((x - 2)(x + 2)ig)^2 = (x^2 - 4)^2 = x^4 - 8x^2 + 16$$. 35. Correct choice is (b). 36. Problem: Simplify $(z^2 + 2z + 1)(z^2 - 2z + 1)$. 37. Recognize each as perfect squares: $$(z^2 + 2z + 1) = (z+1)^2, \quad (z^2 - 2z +1) = (z-1)^2$$. 38. So product is: $$(z+1)^2 (z-1)^2 = ig((z+1)(z-1)ig)^2 = (z^2 -1)^2 = z^4 - 2 z^2 +1$$. 39. Correct choice is (a). 40. Problem: Simplify $(x^2 + 2y)\{(x+y)^2 - (x - y)^2\}$. 41. Expand the squared terms: $$ (x+y)^2 = x^2 + 2xy + y^2$$ $$ (x - y)^2 = x^2 - 2xy + y^2$$ 42. Subtract: $$(x+y)^2 - (x - y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 4xy$$. 43. Multiply by $(x^2 + 2y)$: $$(x^2 + 2y)(4xy) = 4x^3 y + 8 x y^2$$. 44. None of the given options matches, so correct is none. Total questions solved: 9.