Question

If `x^2 + 1/x^2 = 27`, find the value of `x^4 + 1/x^4`.

Answer 1

Given: `x^2 + 1/x^2 = 27`

Step-wise calculation:

1. We want to find `x^4 + 1/x^4`.

2. Use the identity:

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

3. Rearranging to isolate `x^4 + 1/x^4`, we get:

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

4. Substitute `x^2 + 1/x^2 = 27` into the equation:

`x^4 + 1/x^4 = 27^2 - 2`

`x^4 + 1/x^4 = 729 - 2`

`x^4 + 1/x^4 = 727`

So, the value of `x^4 + 1/x^4` is 727.