If x^4 + \frac{1}{x^4} = 727, x > 1, then what is the value of \left(x - \frac{1}{x}\right)?

Question 66

If $$x^4 + \frac{1}{x^4} = 727, x > 1$$, then what is the value of $$\left(x - \frac{1}{x}\right)?$$

Solution

$$x^4+\frac{1}{x^4}=727$$

$$x^4+\frac{1}{x^4}+2=729$$

$$\left(x^2+\frac{1}{x^2}\right)^2=729$$

$$x^2+\frac{1}{x^2}=27$$

$$x^2+\frac{1}{x^2}-2=25$$

$$\left(x-\frac{1}{x}\right)^2=25$$

Since $$x>1$$,

$$x-\frac{1}{x}=5$$

Hence, the correct answer is Option D

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