Simplify the following fractions (assume all denominators are non-zero): \( \begin{array}{ll}\text { a) } \frac{x^{2}-y^{2}}{3} \times \frac{6 x-3 y}{x^{3}+y^{3}} & \text { b) } \frac{2 x^{3}-16 y^{3}}{4 x+8 y} \times \frac{x^{2}-4 y^{2}}{x^{2}+2 x y+4 y^{2}} \\ \text { c) } \frac{x^{2}+5 x+6}{2 x+4} \times \frac{x-3}{x^{2}-9} & \text { d) } \frac{x^{2}+3 x-4}{x^{3}+64 y^{3}} \div \frac{x^{2}-1}{x^{2}-4 x y+16 y^{2}} \\ \text { e) } \frac{x^{2}+4 x-45}{x^{2}-25} \div \frac{x^{2}-5 x+25}{x^{3}+125} & \text { f) } \frac{a x+a y}{x^{2}-y^{2}} \times \frac{x^{2}+2 x y-3 y^{2}}{a} \\ \text { g) } \frac{x^{2}-16}{x^{3}+64} \div \frac{3 x-12}{x^{2}-4 x+16} & \text { h) } \frac{x^{2}-6 x+5}{x^{2}+x+1} \div \frac{3 x-15}{x^{3}-1} \\ \text { i) } \frac{8 x^{3}-27}{4 x^{2}-9} \times \frac{8 x-12}{4 x^{2}+6 x+9} & \text { j) } \frac{x^{2}-5 x+6}{y^{3}} \div \frac{x^{2}-9}{x^{3}}\end{array} \)

Let's simplify each of the provided fractions. a) \[ \frac{x^{2}-y^{2}}{3} \times \frac{6 x-3 y}{x^{3}+y^{3}} = \frac{(x-y)(x+y)}{3} \times \frac{3(2x-y)}{(x+y)(x^2-xy+y^2)} = \frac{(x-y)(2x-y)}{x^2-xy+y^2} \] b) \[ \frac{2 x^{3}-16 y^{3}}{4 x+8 y} \times \frac{x^{2}-4 y^{2}}{x^{2}+2 x y+4 y^{2}} = \frac{2(x^{3}-8y^{3})}{4(x+2y)} \times \frac{(x-2y)(x+2y)}{(x+2y)^{2}} = \frac{(x-2y)(x^{3}-8y^{3})}{2(x+2y)} = \frac{(x-2y)(x-2y)(x+2y)(x^{2}+4y^{2})}{2(x+2y)} \] c) \[ \frac{x^{2}+5 x+6}{2 x+4} \times \frac{x-3}{x^{2}-9} = \frac{(x+2)(x+3)}{2(x+2)} \times \frac{x-3}{(x-3)(x+3)} = \frac{(x+3)}{2} = \frac{(x-3)(x+3)}{2} \] d) \[ \frac{x^{2}+3 x-4}{x^{3}+64 y^{3}} \div \frac{x^{2}-1}{x^{2}-4 x y+16 y^{2}} = \frac{(x+4)(x-1)}{(x+4)(x^{2}-4y^{2})} \div \frac{(x-1)(x+1)}{(x-4)(x-4y)} = \frac{(x-1)}{(x+4)(x+1)} \] e) \[ \frac{x^{2}+4 x-45}{x^{2}-25} \div \frac{x^{2}-5 x+25}{x^{3}+125} = \frac{(x+9)(x-5)}{(x+5)(x-5)} \div \frac{(x-5)(x-5)}{(x+5)(x+5)} = \frac{(x+9)}{(x+5)} \] f) \[ \frac{a x+a y}{x^{2}-y^{2}} \times \frac{x^{2}+2 x y-3 y^{2}}{a} = \frac{a(x+y)}{(x-y)(x+y)} \times \frac{(x-3y)(x+y)}{a} = \frac{(x-3y)}{(x-y)} \] g) \[ \frac{x^{2}-16}{x^{3}+64} \div \frac{3 x-12}{x^{2}-4 x+16} = \frac{x^{2}-16}{(x+4)(x^2-4x+16)} \div \frac{3(x-4)}{(x-4)(x-4)} = \frac{(x+4)(x-4)(x+4)}{(x-4)(x^2-4x+16)} = \frac{(x+4)}{(x+4)} \] h) \[ \frac{x^{2}-6 x+5}{x^{2}+x+1} \div \frac{3 x-15}{x^{3}-1} = \frac{(x-1)(x-5)}{(x^{2}+x+1)} \div \frac{3(x-5)}{(x-1)(x^2+x+16)} = \frac{(x-1)}{3} \] i) \[ \frac{8 x^{3}-27}{4 x^{2}-9} \times \frac{8 x-12}{4 x^{2}+6 x+9} = \frac{(2x-3)(4x^2+6x+9)}{(2x-3)(2x+3)(2x-3