1. **State the problem:** Simplify the expression $$\left(\frac{n^5}{n-6}\right) \cdot \left(\frac{n^2 - 6n}{n^8}\right)$$. 2. **Rewrite the expression:** $$\frac{n^5}{n-6} \times \frac{n^2 - 6n}{n^8}$$ 3. **Factor the numerator in the second fraction:** $$n^2 - 6n = n(n - 6)$$ 4. **Substitute the factorization:** $$\frac{n^5}{n-6} \times \frac{n(n - 6)}{n^8}$$ 5. **Multiply the numerators and denominators:** $$\frac{n^5 \times n(n - 6)}{(n-6) \times n^8} = \frac{n^6 (n - 6)}{(n-6) n^8}$$ 6. **Cancel the common factor $n-6$ in numerator and denominator:** $$\frac{n^6 \cancel{(n - 6)}}{\cancel{(n-6)} n^8} = \frac{n^6}{n^8}$$ 7. **Simplify the powers of $n$:** $$\frac{n^6}{n^8} = n^{6-8} = n^{-2}$$ 8. **Final simplified expression:** $$n^{-2} = \frac{1}{n^2}$$ **Answer:** $$\frac{1}{n^2}$$