Question

If `x - 1/x = 4`, find the value of `x^3 - 1/x^3`.

Answer 1

Given: `x - 1/x = 4`

Step-wise calculation:

1. Square both sides to find `x^2 + 1/x^2`: 

`(x - 1/x)^2 = 4^2`

`x^2 - 2 + 1/x^2 = 16`

`x^2 + 1/x^2 = 16 + 2`

`x^2 + 1/x^2 = 18`

2. Use the identity for the cube difference:

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

3. Substitute the known values:

`x^3 - 1/x^3 = 4 xx (18 + 1)`

`x^3 - 1/x^3 = 4 xx 19`

`x^3 - 1/x^3 = 76`