Simplify the Following
\[ \left(\frac{x}{2}+\frac{y}{3}\right)^3 – \left(\frac{x}{2}-\frac{y}{3}\right)^3 \]
Solution:
Let
\[ a=\frac{x}{2}+\frac{y}{3},\quad b=\frac{x}{2}-\frac{y}{3} \]
Using identity:
\[ a^3-b^3=(a-b)(a^2+ab+b^2) \]
\[ a-b = \frac{2y}{3} \]
\[ a+b = x \]
\[ ab = \left(\frac{x}{2}+\frac{y}{3}\right) \left(\frac{x}{2}-\frac{y}{3}\right) = \frac{x^2}{4}-\frac{y^2}{9} \]
\[ a^2+ab+b^2 = (a+b)^2-ab \]
\[ = x^2-\left(\frac{x^2}{4}-\frac{y^2}{9}\right) \]
\[ = \frac{3x^2}{4}+\frac{y^2}{9} \]
\[ a^3-b^3 = \frac{2y}{3} \left(\frac{3x^2}{4}+\frac{y^2}{9}\right) \]
\[ = \frac{x^2y}{2}+\frac{2y^3}{27} \]