1. **Stating the problem:** Simplify the expression $$\frac{8z}{xy} : \left(\frac{3x + 15}{2x + 10} \cdot \frac{16xz}{3x^2y}\right)$$. 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{8z}{xy} \div \left(\frac{3x + 15}{2x + 10} \cdot \frac{16xz}{3x^2y}\right) = \frac{8z}{xy} \times \frac{1}{\frac{3x + 15}{2x + 10} \cdot \frac{16xz}{3x^2y}}$$ 3. **Simplify the denominator inside the reciprocal:** First, simplify each fraction: - Factor numerator and denominator of $$\frac{3x + 15}{2x + 10}$$: $$3x + 15 = 3(x + 5)$$ $$2x + 10 = 2(x + 5)$$ So, $$\frac{3x + 15}{2x + 10} = \frac{3(x + 5)}{2(x + 5)} = \frac{3}{2}$$ (since $$x + 5 \neq 0$$). 4. **Multiply the simplified fractions in the denominator:** $$\frac{3}{2} \times \frac{16xz}{3x^2y} = \frac{3}{2} \times \frac{16xz}{3x^2y}$$ Cancel 3 in numerator and denominator: $$= \frac{1}{2} \times \frac{16xz}{x^2y} = \frac{16xz}{2x^2y} = \frac{8xz}{x^2y}$$ Cancel $$x$$ in numerator and denominator: $$= \frac{8z}{xy}$$ 5. **Now the original expression becomes:** $$\frac{8z}{xy} \times \frac{1}{\frac{8z}{xy}} = \frac{8z}{xy} \times \frac{xy}{8z}$$ 6. **Simplify the multiplication:** $$= 1$$ (since numerator and denominator are the same and nonzero). **Final answer:** $$1$$