1. **State the problem:** Simplify the expression $$w = \frac{(-2^{-3})^{-3}}{(2^{-2})^{-4}}$$. 2. **Recall the exponent rules:** - Power of a power: $$(a^m)^n = a^{m \times n}$$ - Negative exponent: $$a^{-m} = \frac{1}{a^m}$$ - Division of powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Simplify the numerator:** $$(-2^{-3})^{-3} = (-1 \times 2^{-3})^{-3} = (-1)^{-3} \times (2^{-3})^{-3}$$ 4. Calculate each part: - $$(-1)^{-3} = -1$$ because $(-1)$ raised to an odd power remains $-1$. - $$(2^{-3})^{-3} = 2^{-3 \times -3} = 2^9$$ 5. So numerator becomes: $$-1 \times 2^9 = -2^9$$ 6. **Simplify the denominator:** $$(2^{-2})^{-4} = 2^{-2 \times -4} = 2^8$$ 7. **Rewrite the whole expression:** $$w = \frac{-2^9}{2^8}$$ 8. **Divide powers with the same base:** $$w = -2^{9-8} = -2^1 = -2$$ **Final answer:** $$w = -2$$