Question

Simplify of the following:

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

Answer 1

In the given problem, we have to simplify equation 

Given  `(x+ 2/x)^3 + (x-2/x) `

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

Here  `a= (x+2/x) ,b=(x-2/x)`

By applying identity we get 

` = (x+2/x + x - 2/x-2/x) [(x+2/x)^2 + (x-2/x)^2 - ((x+2/x)  xx (x-2/x))]`

` = (x+2/x + x -2/x) [(x  xx x + 2/x xx 2/x + 2  xx x xx 2/x) +(x  xx x + 2/x xx 2/x - 2  xx x xx 2/x) - (x^2 + 4/x^2)]`

` = (2x)[(x^2 + 4/x^2 +(4x)/x)+ (x^2 + 4/x^2 -(4x)/x) - (x^2 - 4/x^2)]`

` = (2x)[x^2+ 4/x^2 + (4x)/x + x^2 +  4 /x^2 -(4x)/x - x^2 + 4 /x^2]`

By rearranging the variable we get,

` = (2x)[x^2 + 4/x^2 + 4/x^2 + 4/x^2]`

` = 2x xx [x^2+ 12/x^2]`

` = 2x^3 + 24/x`

Hence the simplified value of `(x+2/x)^3+(x-2/x)^3`is `2x^3 + 24/x`.