1. Problem: Simplify the expression $$\left(\frac{8z}{xy}\right) : \left(\frac{3x + 15}{2x + 10} \cdot \frac{16xz}{3x^2y}\right)$$ 2. Recall that division by a fraction is the same as multiplication by its reciprocal. Also, factor expressions where possible. 3. Factor the terms inside the parentheses: - $$3x + 15 = 3(x + 5)$$ - $$2x + 10 = 2(x + 5)$$ 4. Substitute the factored forms: $$\left(\frac{8z}{xy}\right) : \left(\frac{3(x + 5)}{2(x + 5)} \cdot \frac{16xz}{3x^2y}\right)$$ 5. Simplify the product inside the division: $$\frac{3(x + 5)}{2(x + 5)} \cdot \frac{16xz}{3x^2y} = \frac{3}{2} \cdot \frac{16xz}{3x^2y}$$ (since $$x + 5$$ cancels) 6. Simplify the multiplication: $$\frac{3}{2} \cdot \frac{16xz}{3x^2y} = \frac{3 \cdot 16xz}{2 \cdot 3x^2y} = \frac{16xz}{2x^2y}$$ 7. Simplify numerator and denominator: $$\frac{16xz}{2x^2y} = \frac{16}{2} \cdot \frac{xz}{x^2y} = 8 \cdot \frac{z}{xy}$$ 8. Now the original expression becomes: $$\left(\frac{8z}{xy}\right) : \left(8 \cdot \frac{z}{xy}\right)$$ 9. Division by a product is multiplication by reciprocal: $$\frac{8z}{xy} \cdot \frac{1}{8 \cdot \frac{z}{xy}} = \frac{8z}{xy} \cdot \frac{xy}{8z}$$ 10. Simplify: $$\frac{8z}{xy} \cdot \frac{xy}{8z} = 1$$ Final answer: $$1$$