1. **State the problem:** Simplify the expression $$(2 b^{-5})^3$$ and find which of the given options is equivalent for all $b \neq 0$. 2. **Recall the power of a product rule:** $$(xy)^n = x^n y^n$$. This means we raise each factor inside the parentheses to the power outside. 3. **Apply the rule:** $$ (2 b^{-5})^3 = 2^3 (b^{-5})^3 $$ 4. **Calculate powers:** $$ 2^3 = 8 $$ $$ (b^{-5})^3 = b^{-5 \times 3} = b^{-15} $$ 5. **Combine results:** $$ 8 b^{-15} $$ 6. **Rewrite with positive exponents:** $$ 8 b^{-15} = \frac{8}{b^{15}} $$ 7. **Compare with options:** - Option A: $\frac{2}{b^{15}}$ - Option B: $\frac{8}{b^{15}}$ - Option C: $\frac{1}{2 b^{15}}$ - Option D: $\frac{1}{8 b^{15}}$ The expression matches option B. **Final answer:** Option B: $\frac{8}{b^{15}}$