6.43. \(\left(\frac{1}{2}x+\frac{2}{3}y\right)^2\)

\[ \left(\frac{x}{2}+\frac{2y}{3}\right)^2=\frac{x^2}{4}+2\cdot\frac{x}{2}\cdot\frac{2y}{3}+\frac{4y^2}{9}=\frac{x^2}{4}+\frac{2xy}{3}+\frac{4y^2}{9}. \]

Ответ: \(\frac{x^2}{4}+\frac{2xy}{3}+\frac{4y^2}{9}\).