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Question
Find the following product:
\[\left( \frac{x}{2} + 2y \right) \left( \frac{x^2}{4} - xy + 4 y^2 \right)\]
Solution
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`.
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