1. **State the problem:** Simplify the expression $$\frac{(x-8)^3 (x+8)^4}{(x+8)^5 (x-8)^2}$$. 2. **Recall the rule for dividing powers with the same base:** $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Apply the rule to each base separately:** $$\frac{(x-8)^3}{(x-8)^2} = (x-8)^{3-2} = (x-8)^1 = x-8$$ $$\frac{(x+8)^4}{(x+8)^5} = (x+8)^{4-5} = (x+8)^{-1} = \frac{1}{x+8}$$ 4. **Combine the simplified parts:** $$ (x-8) \times \frac{1}{x+8} = \frac{x-8}{x+8} $$ 5. **Final answer:** $$\boxed{\frac{x-8}{x+8}}$$