1. **State the problem:** Identify which expressions are equivalent to $5^5 \cdot 5^{-2}$. 2. **Recall the exponent multiplication rule:** When multiplying powers with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. **Apply the rule:** $$5^5 \cdot 5^{-2} = 5^{5 + (-2)} = 5^3$$ 4. **Check each option:** - $5^{-10}$ is not equal to $5^3$. - $5^3$ is exactly the simplified form. - $\frac{1}{5^{-10}} = 5^{10}$ (since $\frac{1}{a^{-n}} = a^n$), which is not $5^3$. - $\frac{1}{5^3} = 5^{-3}$, which is not $5^3$. 5. **Conclusion:** The only equivalent expression is $5^3$. **Final answer:** $5^3$