1. **Stating the problem:** Simplify the expression $$\frac{\sqrt{18} \times \sqrt{6}}{\sqrt{15} \times \sqrt{5}}$$. 2. **Formula and rules:** Recall that $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$ and $$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$. 3. **Apply the multiplication inside the square roots:** $$\frac{\sqrt{18} \times \sqrt{6}}{\sqrt{15} \times \sqrt{5}} = \frac{\sqrt{18 \times 6}}{\sqrt{15 \times 5}}$$ 4. **Calculate the products inside the roots:** $$18 \times 6 = 108$$ $$15 \times 5 = 75$$ So the expression becomes: $$\frac{\sqrt{108}}{\sqrt{75}} = \sqrt{\frac{108}{75}}$$ 5. **Simplify the fraction inside the root:** $$\frac{108}{75} = \frac{108 \div 3}{75 \div 3} = \frac{36}{25}$$ 6. **Rewrite the expression:** $$\sqrt{\frac{36}{25}} = \frac{\sqrt{36}}{\sqrt{25}}$$ 7. **Calculate the square roots:** $$\sqrt{36} = 6$$ $$\sqrt{25} = 5$$ 8. **Final simplified result:** $$\frac{6}{5}$$ **Answer:** $$\frac{6}{5}$$