Find the following products:$ \left(\frac{x}{2}+2 y\right)\left(\frac{x^{2}}{4}-x y+4 y^{2}\right) $

Given: 

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

To do: 

We have to find the given product.

Solution: 

We know that,

$a^{3}+b^{3}=(a+b)(a^{2}-a b+b^{2})$

$a^{3}-b^{3}=(a-b)(a^{2}+a b+b^{2})$

Therefore,

$(\frac{x}{2}+2 y)(\frac{x^{2}}{4}-x y+4 y^{2})=(\frac{x}{2}+2 y)[(\frac{x}{2})^{2}-\frac{x}{2} \times 2 y+(2 y)^{2}]$

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

$=\frac{x^{3}}{8}+8 y^{3}$

 Hence, $(\frac{x}{2}+2 y)(\frac{x^{2}}{4}-x y+4 y^{2})=\frac{x^{3}}{8}+8 y^{3}$.

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Updated on: 10-Oct-2022

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