Question

If  \[x^2 + \frac{1}{x^2}\], find the value of \[x^3 - \frac{1}{x^3}\]

Answer 1

In the given problem, we have to find the value of  `x^3 - 1/x^3`

Given   `x^2 + 1/x^3 = 51`

We shall use the identity `(x+y)^2 = x^2 + y^2 +2xy`

Here putting `x^2 + 1/x^3 = 51`,

`(x - 1/x)^2 = x^2 +1/x^2 - 2 xx x xx 1/x`

`(x - 1/x)^2 = x^2 +1/x^2 - 2 xx x xx 1/x`

`(x - 1/x)^2 = 51 - 2`

`(x - 1/x)^2 = 49`

`(x - 1/x) = sqrt49`

`(x - 1/x) =±7`

In order to find `x^3 - 1/x^3`we are using identity `a^3 - b^3 = (a-b)(a^2 +b^2 +ab)`

`x^3 - 1/x^3 = (x-1/x)(x^2 + 1/x^2 + x xx 1/x)`

`x^3 - 1/x^3 = (x-1/x)(x^2 + 1/x^2 + x xx 1/x)`

Here  `(x-1/x)= 7`and  `x^2 + 1/x^2 = 51` 

`= 7 (51 +1)`

` = 7 xx 52`

 ` = 364`

Hence the value of  `x^3 - 1/x^3`is  364.