1. **State the problem:** Simplify the expression $$\frac{5^5 z^{-5}}{5^6 z^{-8}}$$. 2. **Recall the laws of exponents:** - When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$. 3. **Apply the exponent rule to the base 5:** $$\frac{5^5}{5^6} = 5^{5-6} = 5^{-1}$$. 4. **Apply the exponent rule to the base z:** $$\frac{z^{-5}}{z^{-8}} = z^{-5 - (-8)} = z^{-5 + 8} = z^3$$. 5. **Combine the results:** $$5^{-1} z^3$$. 6. **Rewrite with positive exponents:** $$5^{-1} = \frac{1}{5}$$, so the expression becomes $$\frac{z^3}{5}$$. **Final answer:** $$\frac{z^3}{5}$$