Find the Cube of the Following Binomial Expression
\[ \frac{3}{x}-\frac{2}{x^2} \]
Solution:
Using identity:
\[ (a-b)^3 = a^3-b^3-3ab(a-b) \]
\[ \left(\frac{3}{x}-\frac{2}{x^2}\right)^3 \]
\[ = \left(\frac{3}{x}\right)^3 – \left(\frac{2}{x^2}\right)^3 – 3\left(\frac{3}{x}\right)\left(\frac{2}{x^2}\right) \left(\frac{3}{x}-\frac{2}{x^2}\right) \]
\[ = \frac{27}{x^3} – \frac{8}{x^6} – \frac{18}{x^3} \left(\frac{3}{x}-\frac{2}{x^2}\right) \]
\[ = \frac{27}{x^3} – \frac{8}{x^6} – \frac{54}{x^4} + \frac{36}{x^5} \]