Question

If x = 3 and y = − 1, find the values of the following using in identify:

\[\left( \frac{3}{x} - \frac{x}{3} \right) \left( \frac{x^2}{9} + \frac{9}{x^2} + 1 \right)\]

Answer 1

In the given problem, we have to find the value of equation using identity

Given \[\left( \frac{3}{x} - \frac{x}{3} \right) \left( \frac{x^2}{9} + \frac{9}{x^2} + 1 \right)\]

We shall use the identity  `(a-b)(a^2 + ab + b^2) = a^3 - b^3`

We can rearrange the  \[\left( \frac{3}{x} - \frac{x}{3} \right) \left( \frac{x^2}{9} + \frac{9}{x^2} + 1 \right)\]as

`= (3/x - x/3) ((3/x)^2 + (x/3)^2 + (3/x)(x/3))`

` = (3/x)^3 - (x/3)^3`

\[= \left( \frac{3}{x} \right) \times \left( \frac{3}{x} \right) \times \left( \frac{3}{x} \right) - \left( \frac{x}{3} \right) \times \left( \frac{x}{3} \right) \times \left( \frac{x}{3} \right)\]

\[ = \frac{27}{x^3} - \frac{x^3}{27}\]

Now substituting the value x=3, in `27/x^3 - x^3/27`we get,

 `27/3^3 - 3^3/27`

 `27/27 - 27/27`

` = 0`

Hence the Product value of  \[\left( \frac{3}{x} - \frac{x}{3} \right) \left( \frac{x^2}{9} + \frac{9}{x^2} + 1 \right)\] is `0`.