Question:
If \[ x=3 \] find the value of:
\[ \left(\frac{3}{x} – \frac{x}{3}\right) \left(\frac{x^2}{9} + \frac{9}{x^2} + 1\right) \]
Solution:
Rearranging the terms:
\[ \left(\frac{3}{x} – \frac{x}{3}\right) \left(\frac{9}{x^2} + \frac{3}{x}\cdot\frac{x}{3} + \frac{x^2}{9}\right) \]
Using identity:
\[ (a-b)(a^2+ab+b^2)=a^3-b^3 \]
Here, \[ a=\frac{3}{x},\qquad b=\frac{x}{3} \]
\[ = \left(\frac{3}{x}\right)^3 – \left(\frac{x}{3}\right)^3 \]
Substituting \[ x=3 \]
\[ = \left(\frac{3}{3}\right)^3 – \left(\frac{3}{3}\right)^3 \]
\[ = 1^3-1^3 \]
\[ =0 \]