Question

Find the following product:

\[\left( \frac{x}{2} + 2y \right) \left( \frac{x^2}{4} - xy + 4 y^2 \right)\]

Answer 1

Given  \[\left( \frac{x}{2} + 2y \right) \left( \frac{x^2}{4} - xy + 4 y^2 \right)\]

We shall use the identity  `a^3 + b^3 = (a+b)(a^2 - ab + b^2)`

We can rearrange the  `(x/2 + 2y) (x^2/4 - xy + 4y^2)`as

` = (x/2 + 2y)[(x/2)^2 - (x/2)(2y)+ (2y)^2]`

` = (x/2)^3 + (2y)^3`

 `=  (x/2 ) xx (x/2 )xx (x/2 )+ (2y) xx (2y) xx (2y) `

`=  x^3/8 + 8y^3`

Hence the Product value of `(x/2 + 2y) (x^2/4 - xy + 4y^2)`is `x^2 / 8 + 8y^3`.