1. **State the problem:** Simplify the rational expression $ \frac{70m^7n^2 - 50m^5n + 10m^3n^4}{10m^2n} $. 2. **Write the formula:** To simplify, divide each term in the numerator by the denominator separately: $$ \frac{a + b + c}{d} = \frac{a}{d} + \frac{b}{d} + \frac{c}{d} $$ 3. **Divide each term:** $$ \frac{70m^7n^2}{10m^2n} - \frac{50m^5n}{10m^2n} + \frac{10m^3n^4}{10m^2n} $$ 4. **Simplify coefficients:** $$ \frac{\cancel{70}^{7}m^7n^2}{\cancel{10}^{1}m^2n} = 7 \cdot \frac{m^7}{m^2} \cdot \frac{n^2}{n} $$ $$ \frac{\cancel{50}^{5}m^5n}{\cancel{10}^{1}m^2n} = 5 \cdot \frac{m^5}{m^2} \cdot \frac{n}{n} $$ $$ \frac{\cancel{10}^{1}m^3n^4}{\cancel{10}^{1}m^2n} = 1 \cdot \frac{m^3}{m^2} \cdot \frac{n^4}{n} $$ 5. **Simplify powers:** $$ m^{7-2} = m^5, \quad n^{2-1} = n^1 = n $$ $$ m^{5-2} = m^3, \quad n^{1-1} = n^0 = 1 $$ $$ m^{3-2} = m^1 = m, \quad n^{4-1} = n^3 $$ 6. **Write the simplified terms:** $$ 7m^5n - 5m^3 + mn^3 $$ 7. **Final answer:** $$ \boxed{7m^5n - 5m^3 + mn^3} $$