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If $x = 3$ and $y = -1$, find the values of each of the following using in identity:
\( \left(\frac{3}{x}-\frac{x}{3}\right)\left(\frac{x^{2}}{9}+\frac{9}{x^{2}}+1\right) \)
Given:
$x = 3$ and $y = -1$
To do:
We have to find the value of \( \left(\frac{3}{x}-\frac{x}{3}\right)\left(\frac{x^{2}}{9}+\frac{9}{x^{2}}+1\right) \).
Solution:
We know that,
$a^{3}+b^{3}=(a+b)(a^{2}-a b+b^{2})$
$a^{3}-b^{3}=(a-b)(a^{2}+a b+b^{2})$
Therefore,
$(\frac{3}{x}-\frac{x}{3})(\frac{x^{2}}{9}+\frac{9}{x^{2}}+1)=(\frac{3}{x}-\frac{x}{3})[(\frac{x}{3})^{2}+\frac{x}{3} \times \frac{3}{x}+(\frac{3}{x})^{2}]$
$=(\frac{3}{x})^{3}-(\frac{x}{3})^{3}$
$=(\frac{3}{3})^{3}-(\frac{3}{3})^{3}$
$=1^{3}-1^{3}$
$=0$
Hence, $(\frac{3}{x}-\frac{x}{3})(\frac{x^{2}}{9}+\frac{9}{x^{2}}+1)=0$.