1. **State the problem:** Simplify the expression $\left(\frac{2m^3}{n^2}\right)^2$. 2. **Recall the rule:** When raising a fraction to a power, raise both numerator and denominator to that power: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$. 3. **Apply the power to numerator and denominator:** $$\left(\frac{2m^3}{n^2}\right)^2 = \frac{(2m^3)^2}{(n^2)^2}$$ 4. **Simplify numerator:** $$(2m^3)^2 = 2^2 \cdot (m^3)^2 = 4m^{3 \times 2} = 4m^6$$ 5. **Simplify denominator:** $$(n^2)^2 = n^{2 \times 2} = n^4$$ 6. **Write the simplified expression:** $$\frac{4m^6}{n^4}$$ **Final answer:** $\boxed{\frac{4m^6}{n^4}}$