If \(x^4+\frac{1}{x^4}=322\) and x > 1, what is the value of the expression \(x-\frac{1}{x}\)?

Solution

Solution: Use \((x^2+\frac1{x^2})^2=x^4+\frac1{x^4}+2\). Therefore, \((x^2+\frac1{x^2})^2=322+2=324\), giving \(x^2+\frac1{x^2}=18\). Now, \((x-\frac1x)^2=x^2+\frac1{x^2}-2=18-2=16\). Since x > 1, \(x-\frac1x\) is positive, so the value is 4. Trick: Convert the fourth-power expression into a square, then use the identity for the square of a difference.