Question

Find the following product:

$$(\frac{2}{x}+3x)(\frac{4}{x^2}+9x^2-6)$$

Answer 1

Given  ${\displaystyle (\frac{2}{x}+3x)(\frac{4}{x^2}+9x^2-6)}$

We shall use the identity, ${\displaystyle a^3+b^3=(a+b)(a^2+b^2-ab)}$

We can rearrange the ${\displaystyle (\frac{2}{x}+3x)(\frac{4}{x^3}+9x^2-6)}$as

${\displaystyle =(\frac{2}{x}+3x)[{\left(\frac{2}{x}\right)}^2+{\left(3x\right)}^2-\left(\frac{2}{x}\right)\left(3x\right)]}$

${\displaystyle =\left(\frac{2}{x^3}\right)+{\left(3x\right)}^3}$

${\displaystyle =\left(\frac{2}{x}\right)\times \left(\frac{2}{x}\right) \times \left(\frac{2}{x}\right)+\left(3x\right)\times \left(3x\right)\times \left(3x\right)}$

${\displaystyle =\frac{8}{x^3}+27x^3}$

Hence the Product value of  ${\displaystyle (\frac{2}{x}+3x)(\frac{4}{x^2}+9x^2-6)}$is  ${\displaystyle \frac{8}{x^3}+27x^3}$.