1. **State the problem:** Simplify the expression $$\left(\frac{2n^4}{3m^3n^7}\right)^2$$ and write the answer using only positive exponents. 2. **Recall the formula:** When raising a fraction to a power, raise both numerator and denominator to that power: $$\left(\frac{a}{b}\right)^c = \frac{a^c}{b^c}$$ 3. **Apply the power to numerator and denominator:** $$\left(\frac{2n^4}{3m^3n^7}\right)^2 = \frac{(2n^4)^2}{(3m^3n^7)^2}$$ 4. **Simplify numerator and denominator separately:** $$ (2n^4)^2 = 2^2 \cdot (n^4)^2 = 4n^{8} $$ $$ (3m^3n^7)^2 = 3^2 \cdot (m^3)^2 \cdot (n^7)^2 = 9m^{6}n^{14} $$ 5. **Rewrite the fraction:** $$ \frac{4n^{8}}{9m^{6}n^{14}} $$ 6. **Simplify the powers of $n$ by subtracting exponents:** $$ \frac{4\cancel{n^{8}}}{9m^{6}n^{14}} = \frac{4}{9m^{6}n^{14-8}} = \frac{4}{9m^{6}n^{6}} $$ 7. **Final simplified expression with positive exponents:** $$ \boxed{\frac{4}{9m^{6}n^{6}}} $$