Question

If \[x - \frac{1}{x} = 5\], 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 - 1/x = 5`,

We shall use the identity (a − b)³ = a³ − b³ − 3ab(a − b),

Here putting, `x- 1/x = 5`,

`(x-1/x)^3 = x^3 - 1/x^3 - 3(x xx1/x)(x - 1/x)`

`(5)^3 = x^3 - 1/x^3 - 3(x xx1/x)(x - 1/x)`

`125 = x^3 - 1/x^3 - 3(x - 1/x)`

`125 = x^3 - 1/x^3 - 3xx5`

`125 = x^3 - 1/x^3 - 15`

`125 + 15 = x^3 - 1/x^3`

`140 = x^3 - 1/x^3`

Hence, the value of `x^3 - 1/x^3`  is 140.