1. **Problem:** Simplify $\frac{5y^3}{32x} \times \frac{-4}{15x^2 y^3}$. 2. **Formula:** Multiply numerators and denominators, then simplify common factors. 3. **Work:** Numerator: $5y^3 \times (-4) = -20y^3$. Denominator: $32x \times 15x^2 y^3 = 480x^3 y^3$. 4. Cancel $y^3$ from numerator and denominator: $\frac{-20}{480x^3} = \frac{-1}{24x^3}$. --- 1. **Problem:** Simplify $\frac{y^3}{8} \times \frac{9x^2}{y^3}$. 2. Multiply numerators and denominators: Numerator: $y^3 \times 9x^2 = 9x^2 y^3$. Denominator: $8 \times y^3 = 8 y^3$. 3. Cancel $y^3$: $\frac{9x^2}{8}$. --- 1. **Problem:** Simplify $\frac{1}{4x-3} \times (20x-15)$. 2. Factor $20x-15 = 5(4x-3)$. 3. Multiply: $\frac{1}{4x-3} \times 5(4x-3) = 5$. --- 1. **Problem:** Simplify $\frac{x-6}{2x+5} \times \frac{2x}{-x+6}$. 2. Note $-x+6 = -(x-6)$. 3. Multiply: $\frac{x-6}{2x+5} \times \frac{2x}{-(x-6)} = \frac{x-6}{2x+5} \times \frac{2x}{-(x-6)}$. 4. Cancel $x-6$: $\frac{1}{2x+5} \times \frac{2x}{-1} = \frac{-2x}{2x+5}$. --- 1. **Problem:** Simplify $\frac{x^3 + x}{5} \times \frac{10}{x^2 + x}$. 2. Factor: $x^3 + x = x(x^2 + 1)$ (cannot factor further), $x^2 + x = x(x+1)$. 3. Multiply: $\frac{x(x^2 + 1)}{5} \times \frac{10}{x(x+1)}$. 4. Cancel $x$: $\frac{x^2 + 1}{5} \times \frac{10}{x+1} = \frac{10(x^2 + 1)}{5(x+1)} = \frac{2(x^2 + 1)}{x+1}$. --- 1. **Problem:** Simplify $\frac{x^2 + 6x + 5}{x} \times \frac{x^4}{3x + 3}$. 2. Factor: $x^2 + 6x + 5 = (x+5)(x+1)$, $3x + 3 = 3(x+1)$. 3. Multiply: $\frac{(x+5)(x+1)}{x} \times \frac{x^4}{3(x+1)}$. 4. Cancel $x+1$: $\frac{x+5}{x} \times \frac{x^4}{3} = \frac{(x+5)x^4}{3x} = \frac{(x+5)x^3}{3}$. --- 1. **Problem:** Simplify $\frac{x^2 - 1}{(x - 1)^2} \times \frac{x - 1}{x^2 + 2x + 1}$. 2. Factor: $x^2 - 1 = (x-1)(x+1)$, $x^2 + 2x + 1 = (x+1)^2$. 3. Multiply: $\frac{(x-1)(x+1)}{(x-1)^2} \times \frac{x-1}{(x+1)^2}$. 4. Cancel $x-1$ and $x+1$: $\frac{(x+1)}{(x-1)} \times \frac{1}{(x+1)} = \frac{1}{x-1}$. --- 1. **Problem:** Simplify $\frac{x^2 - 2x}{xy - 2y + 3x - 6} \times \frac{8y + 24}{3x + 6}$. 2. Factor: $x^2 - 2x = x(x-2)$, $xy - 2y + 3x - 6 = y(x-2) + 3(x-2) = (x-2)(y+3)$, $8y + 24 = 8(y+3)$, $3x + 6 = 3(x+2)$. 3. Multiply: $\frac{x(x-2)}{(x-2)(y+3)} \times \frac{8(y+3)}{3(x+2)}$. 4. Cancel $(x-2)$ and $(y+3)$: $\frac{x}{1} \times \frac{8}{3(x+2)} = \frac{8x}{3(x+2)}$. --- 1. **Problem:** Simplify $\frac{x}{x^2 - y^2} \times \frac{x + y}{x^2 + xy}$. 2. Factor: $x^2 - y^2 = (x - y)(x + y)$, $x^2 + xy = x(x + y)$. 3. Multiply: $\frac{x}{(x - y)(x + y)} \times \frac{x + y}{x(x + y)}$. 4. Cancel $x$ and $x + y$: $\frac{1}{x - y}$. --- 1. **Problem:** Simplify $\frac{2x^2 - 9x + 9}{8x - 12} \times \frac{2x}{x^2 - 3x}$. 2. Factor: $8x - 12 = 4(2x - 3)$, $x^2 - 3x = x(x - 3)$. 3. Factor numerator $2x^2 - 9x + 9$: Try factors of $2x^2 - 9x + 9$: $(2x - 3)(x - 3) = 2x^2 - 6x - 3x + 9 = 2x^2 - 9x + 9$. 4. Multiply: $\frac{(2x - 3)(x - 3)}{4(2x - 3)} \times \frac{2x}{x(x - 3)}$. 5. Cancel $(2x - 3)$ and $(x - 3)$ and $x$: $\frac{1}{4} \times 2 = \frac{1}{2}$. **Final answers:** 1. $\frac{-1}{24x^3}$ 2. $\frac{9x^2}{8}$ 3. $5$ 4. $\frac{-2x}{2x+5}$ 5. $\frac{2(x^2 + 1)}{x+1}$ 6. $\frac{(x+5)x^3}{3}$ 7. $\frac{1}{x-1}$ 8. $\frac{8x}{3(x+2)}$ 9. $\frac{1}{x - y}$ 10. $\frac{1}{2}$