Question

If `x - 2/x = 2`, find the value of `x^3 - 8/x^3`.

Answer 1

Given: `x - 2/x = 2`

Find the value of `x^3 - 8/x^3`

Stepwise calculation:

1. Cube both sides of the given equation:

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

`(x - 2/x)^3 = 8`

2. Expand the left side using the identity:

(a – b)³ = a³ – b³ – 3ab(a – b)

Here, `a = x` and `b = 2/x`, 

So, `(x - 2/x)^3 = x^3 - (2/x)^3 - 3 xx x xx 2/x (x - 2/x)`

Simplify:

= `x^3 - 8/x^3 - 3 xx 2 xx (x - 2/x)`

= `x^3 - 8/x^3 - 6(x - 2/x)`

3. Substitute the given value `x - 2/x = 2`:

`8 = x^3 - 8/x^3 - 6 xx 2`

`8 = x^3 - 8/x^3 - 12`

4. Rearrange to solve for `x^3 - 8/x^3`:

`x^3 - 8/x^3 = 8 + 12`

`x^3 - 8/x^3 = 20`

So, the value of `x^3 - 8/x^3` is 20.