$16(x^4 + \frac{1}{x^4}) - 257 = 0$
\(\frac{17}{4}\)
To solve the problem, we need to find the value of \(x^2 + \frac{1}{x^2}\) given the equation:
\(16(x^4 + \frac{1}{x^4}) - 257 = 0.\)
Let's proceed step-by-step:
- \(16(x^4 + \frac{1}{x^4}) = 257\)
- Simplifying, we get:
- \(x^4 + \frac{1}{x^4} = \frac{257}{16}\)
- Recall the identity:
- \((x^2 + \frac{1}{x^2})^2 = x^4 + \frac{1}{x^4} + 2\)
- Now, substitute the value we found:
- \((x^2 + \frac{1}{x^2})^2 = \frac{257}{16} + 2\)
- Convert and simplify:
- \(\frac{257}{16} + 2 = \frac{257}{16} + \frac{32}{16} = \frac{257 + 32}{16} = \frac{289}{16}\)
- Thus,
- \((x^2 + \frac{1}{x^2})^2 = \frac{289}{16}\)
- Taking the square root on both sides gives:
- \(x^2 + \frac{1}{x^2} = \sqrt{\frac{289}{16}} = \frac{\sqrt{289}}{\sqrt{16}} = \frac{17}{4}\)
Finally, the correct answer to the problem according to computations and given options is \(\frac{17}{4}\)
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