1. **State the problem:** Simplify the expression $\left(\frac{1}{5^2}\right)^3$. 2. **Recall the exponent rule:** When raising a power to another power, multiply the exponents: $\left(a^m\right)^n = a^{m \times n}$. 3. **Apply the rule to the base:** Here, the base is $\frac{1}{5^2} = 5^{-2}$, so $$\left(5^{-2}\right)^3 = 5^{-2 \times 3} = 5^{-6}.$$ 4. **Rewrite the negative exponent:** $5^{-6} = \frac{1}{5^6}$. 5. **Calculate $5^6$:** $$5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625.$$ 6. **Final answer:** $$\left(\frac{1}{5^2}\right)^3 = \frac{1}{15625}.$$