1. **State the problem:** Simplify the expression $(-5^5)^2$. 2. **Recall the order of operations:** Exponents are evaluated before applying the negative sign if the negative is not inside parentheses. 3. **Evaluate the inner exponent:** Calculate $5^5$. $$5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125$$ 4. **Apply the negative sign:** Since the negative is outside the exponent, it applies after calculating $5^5$. $$-5^5 = -3125$$ 5. **Square the result:** Now square $-3125$. $$(-3125)^2 = (-3125) \times (-3125)$$ 6. **Calculate the square:** $$(-3125) \times (-3125) = 3125^2 = 9765625$$ 7. **Final answer:** $$(-5^5)^2 = 9765625$$