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प्रश्न
Find the following product:
\[\left( \frac{3}{x} - \frac{5}{y} \right) \left( \frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy} \right)\]
उत्तर
Given \[\left( \frac{3}{x} - \frac{5}{y} \right) \left( \frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy} \right)\]
We shall use the identity `(a-b)(a^2 + ab+ b^2) = a^3 - b^3`
We can rearrange the ` (3/5 - 5/y) (9^2/x^2 + 25/y^2 + 15/(xy))`as
` = ((3/x - 5/y) ((3/x)^2 + (5/y)^2 + (3/x)(5/y))`
` = (3/x)^3 - (5/y)^3 `
\[= \left( \frac{3}{x} \right) \times \left( \frac{3}{x} \right) \times \left( \frac{3}{x} \right) - \left( \frac{5}{y} \right) \times \left( \frac{5}{y} \right) \times \left( \frac{5}{y} \right)\]
\[ = \frac{27}{x^3} - \frac{125}{y^3}\]
Hence the Product value of `(3/x - 5/y) (9^2/x^2 + 25/y^3 + 15/(xy))`is `27/x^3 - 125/y^3`.
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