Question

Find the following product:

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

Answer 1

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`.