Question

If `x - 1/x = 7`, find the value of `x^4 + 1/x^4`.

Answer 1

Given: `x - 1/x = 7`

Step-wise calculation:

1. Square both sides:

`(x - 1/x)^2 = 7^2`

`x^2 - 2 + 1/x^2 = 49`

`x^2 + 1/x^2 - 2 = 49`

`x^2 + 1/x^2 = 51`

2. Square `x^2 + 1/x^2` to find `x^4 + 1/x^4`:

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

`51^2 = x^4 + 2 + 1/x^4` 

`2601 = x^4 + 1/x^4 + 2`

3. Solve for `x^4 + 1/x^4`: 

`x^4 + 1/x^4 = 2601 - 2`

`x^4 + 1/x^4 = 2599`

Therefore, the value of `x^4 + 1/x^4` given `x - 1/x = 7` is 2599.