1. **State the problem:** Simplify the expression $$(4z^{-5})^3$$. 2. **Recall the power of a product rule:** When raising a product to a power, raise each factor to that power separately: $$ (ab)^n = a^n b^n $$ 3. **Apply the rule:** $$ (4z^{-5})^3 = 4^3 (z^{-5})^3 $$ 4. **Calculate each part:** $$ 4^3 = 64 $$ $$ (z^{-5})^3 = z^{-5 \times 3} = z^{-15} $$ 5. **Combine the results:** $$ 64 z^{-15} $$ 6. **Rewrite with positive exponents:** $$ 64 \frac{1}{z^{15}} = \frac{64}{z^{15}} $$ **Final answer:** $$ \frac{64}{z^{15}} $$