Find the Product:
\[ \left(\frac{3}{x} – \frac{5}{y}\right) \left(\frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy}\right) \]
Solution:
Using identity:
\[ (a-b)(a^2+ab+b^2)=a^3-b^3 \]
Here, \[ a=\frac{3}{x},\qquad b=\frac{5}{y} \]
\[ \left(\frac{3}{x} – \frac{5}{y}\right) \left(\frac{9}{x^2} + \frac{15}{xy} + \frac{25}{y^2}\right) \]
\[ = \left(\frac{3}{x}\right)^3 – \left(\frac{5}{y}\right)^3 \]
\[ = \frac{27}{x^3} – \frac{125}{y^3} \]