Find the Product:
\[ \left(\frac{x}{2} + 2y\right)\left(\frac{x^2}{4} – xy + 4y^2\right) \]
Solution:
Using identity:
\[ (a+b)(a^2-ab+b^2)=a^3+b^3 \]
Here, \[ a=\frac{x}{2},\qquad b=2y \]
\[ \left(\frac{x}{2} + 2y\right)\left(\frac{x^2}{4} – xy + 4y^2\right) \]
\[ = \left(\frac{x}{2} + 2y\right) \left[\left(\frac{x}{2}\right)^2 – \left(\frac{x}{2}\right)(2y) + (2y)^2\right] \]
\[ = \left(\frac{x}{2}\right)^3 + (2y)^3 \]
\[ = \frac{x^3}{8} + 8y^3 \]