Find the following products:$ \left(\frac{3}{x}-\frac{5}{y}\right)\left(\frac{9}{x^{2}}+\frac{25}{y^{2}}+\frac{15}{x y}\right) $

Given: 

\( \left(\frac{3}{x}-\frac{5}{y}\right)\left(\frac{9}{x^{2}}+\frac{25}{y^{2}}+\frac{15}{x y}\right) \)

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,

$(\frac{3}{x}-\frac{5}{y})(\frac{9}{x^{2}}+\frac{25}{y^{2}}+\frac{15}{x y})=(\frac{3}{x}-\frac{5}{y})[(\frac{3}{x})^{2}+\frac{3}{x} \times \frac{5}{y}+(\frac{5}{y})^{2}]$

$=(\frac{3}{x})^{3}-(\frac{5}{y})^{3}$

$=\frac{27}{x^{3}}-\frac{125}{y^{3}}$

 Hence, $(\frac{3}{x}-\frac{5}{y})(\frac{9}{x^{2}}+\frac{25}{y^{2}}+\frac{15}{x y})=\frac{27}{x^{3}}-\frac{125}{y^{3}}$.

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Updated on: 10-Oct-2022

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