Question

Expand:${(x-\frac{2}{3}y)}^3$

Answer 1

${(x-\frac{2y}{3})}^3$ is of the form ${(a-b)}^3=a^3-b^3-3ab(a-b)$ where $a=x,b=\frac{2y}{3}$

$={\left(x\right)}^3-{\left(\frac{2y}{3}\right)}^3-3\left(x\right)\left(\frac{2y}{3}\right)(x-\frac{2y}{3})$

$=x^3-\frac{8y^3}{27}-2xy(x-\frac{2y}{3})$

$=x^3-\frac{8y^3}{27}-2x^{2}y+\frac{4x{y}^2}{3}$

$=\frac{1}{27}(27x^3-8y^3-54x^{2}y+36x{y}^2)$

Hence, ${(x-\frac{2}{3}y)}^3=\frac{1}{27}(27x^3-8y^3-54x^{2}y+36x{y}^2)$