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Find the following products:
\( \left(\frac{2}{x}+3 x\right)\left(\frac{4}{x^{2}}+9 x^{2}-6\right) \)
Given:
\( \left(\frac{2}{x}+3 x\right)\left(\frac{4}{x^{2}}+9 x^{2}-6\right) \)
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,
$(\frac{2}{x}+3 x)(\frac{4}{x^{2}}+9 x^{2}-6)=(\frac{2}{x}+3 x)(\frac{4}{x^{2}}-6+9 x^{2})$
$=(\frac{2}{x}+3 x)[(\frac{2}{x})^{2}-\frac{2}{x} \times 3 x+(3 x)^{2}]$
$=(\frac{2}{x})^{3}+(3 x)^{3}$
$=\frac{8}{x^{3}}+27 x^{3}$
Hence, $(\frac{2}{x}+3 x)(\frac{4}{x^{2}}+9 x^{2}-6)=\frac{8}{x^{3}}+27 x^{3}$.
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