Question

Simplify
${(\frac{x}{3}+\frac{y}{5})}^3-{(\frac{x}{3}-\frac{y}{5})}^3$

Answer 1

${(\frac{x}{3}+\frac{y}{5})}^3-$ ${(\frac{x}{3}-\frac{y}{5})}^2$ is of the form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$=(\frac{x}{3}+\frac{y}{5}-\frac{x}{3}+\frac{y}{5})({(\frac{x}{3}+\frac{y}{5})}^2+(\frac{x}{3}+\frac{y}{5})(\frac{x}{3}-\frac{y}{5})+{(\frac{x}{3}-\frac{y}{5})}^2)$

$=\frac{2y}{5}(\frac{x^2}{9}+\frac{y^2}{25}+\frac{2xy}{15}+\frac{x^2}{9}-\frac{y^2}{25}+\frac{x^2}{9}+\frac{y^2}{25}-\frac{2xy}{15})$

$=\frac{2y}{5}(\frac{3x^2}{9}+\frac{y^2}{25})$

$=\left(\frac{2y}{5}\right)(\frac{x^2}{3}+\frac{y^2}{25})$.