If x4 + 1/x4 = 322, x ≠ 0, then one of the values of (x - 1/x) is

For any nonzero x, the difference and sum x − 1/x and x + 1/x connect to x2 + 1/x2 through (x − 1/x)2 = x2 + 1/x2 − 2, and x2 + 1/x2 connects to x4 + 1/x4 through (x2 + 1/x2)2 = x4 + 1/x4 + 2. Squaring each identity in turn lets a fourth-power sum be traced back to the value of x − 1/x without solving for x itself.

1. Let m = x2 + 1/x2. Squaring m gives m2 = (x2 + 1/x2)2 = x4 + 1/x4 + 2.

2. Substituting the given x4 + 1/x4 = 322: m2 = 322 + 2 = 324, so m = √324 = 18 (the positive root, since x2 + 1/x2 ≥ 2 for every real nonzero x).

3. Apply the same squaring identity to x − 1/x: (x − 1/x)2 = x2 + 1/x2 − 2 = m − 2 = 18 − 2 = 16.

4. Taking the square root: x − 1/x = ±√16 = ±4.

Checking numerically: x2 + 1/x2 = 18 gives x2 ≈ 17.944 (the larger root of x4 − 18x2 + 1 = 0), so x ≈ 4.236 and 1/x ≈ 0.236. Then x − 1/x ≈ 4.000 and x4 ≈ 322.0, matching the given equation.

So one value of x − 1/x is 4.