1. **State the problem:** Simplify the expression $$\frac{4}{3} m^3 - \left(\frac{1}{2} m\right)^2 + \left(\frac{3}{5} mn\right)^2 - \left(\frac{1}{4} n^3\right) \left(\frac{2}{3} m^2 + \frac{3}{5} n^2 - \frac{2}{3} mn\right)$$. 2. **Recall the rules:** - Square of a product: $\left(\frac{a}{b} x\right)^2 = \frac{a^2}{b^2} x^2$ - Distributive property: $a(b+c) = ab + ac$ - Multiplying powers with variables: multiply coefficients and add exponents of like variables. 3. **Calculate each term:** - $\left(\frac{1}{2} m\right)^2 = \frac{1^2}{2^2} m^2 = \frac{1}{4} m^2$ - $\left(\frac{3}{5} mn\right)^2 = \frac{3^2}{5^2} m^2 n^2 = \frac{9}{25} m^2 n^2$ 4. **Expand the product:** $$\left(\frac{1}{4} n^3\right) \left(\frac{2}{3} m^2 + \frac{3}{5} n^2 - \frac{2}{3} mn\right) = \frac{1}{4} n^3 \cdot \frac{2}{3} m^2 + \frac{1}{4} n^3 \cdot \frac{3}{5} n^2 - \frac{1}{4} n^3 \cdot \frac{2}{3} mn$$ Calculate each: - $\frac{1}{4} \cdot \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$, so $\frac{1}{6} m^2 n^3$ - $\frac{1}{4} \cdot \frac{3}{5} = \frac{3}{20}$, so $\frac{3}{20} n^{5}$ - $\frac{1}{4} \cdot \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$, so $\frac{1}{6} m n^{4}$ 5. **Rewrite the expression:** $$\frac{4}{3} m^3 - \frac{1}{4} m^2 + \frac{9}{25} m^2 n^2 - \left( \frac{1}{6} m^2 n^3 + \frac{3}{20} n^5 - \frac{1}{6} m n^4 \right)$$ 6. **Distribute the minus sign:** $$\frac{4}{3} m^3 - \frac{1}{4} m^2 + \frac{9}{25} m^2 n^2 - \frac{1}{6} m^2 n^3 - \frac{3}{20} n^5 + \frac{1}{6} m n^4$$ 7. **Final simplified expression:** $$\boxed{\frac{4}{3} m^3 - \frac{1}{4} m^2 + \frac{9}{25} m^2 n^2 - \frac{1}{6} m^2 n^3 - \frac{3}{20} n^5 + \frac{1}{6} m n^4}$$ This is the simplified form; no like terms can be combined further due to different powers of $m$ and $n$.