1. **Stating the problem:** Simplify the expression $$\frac{3^{2} \cdot 7^{3} \cdot x}{7^{8} \cdot 1}$$. 2. **Formula and rules:** When dividing powers with the same base, subtract the exponents: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$. 3. **Apply the rule to the base 7:** $$\frac{7^{3}}{7^{8}} = 7^{3-8} = 7^{-5}$$ 4. **Rewrite the expression:** $$3^{2} \cdot 7^{-5} \cdot x$$ 5. **Calculate $3^{2}$:** $$3^{2} = 9$$ 6. **Final simplified expression:** $$9 \cdot 7^{-5} \cdot x = \frac{9x}{7^{5}}$$ **Answer:** $$\frac{9x}{7^{5}}$$