$16(x^4 + \frac{1}{x^4}) - 257 = 0$
\(\frac{65}{8}\)
To find the value of \(x^3 + \frac{1}{x^3}\), we have been given the equation:
\(16\left(x^4 + \frac{1}{x^4}\right) - 257 = 0\)
Let's work through the solution to find the required expression:
- Firstly, solve the given equation for \(x^4 + \frac{1}{x^4}\):
\(16(x^4 + \frac{1}{x^4}) = 257\)
- Divide both sides by 16 to isolate \(x^4 + \frac{1}{x^4}\):
\(x^4 + \frac{1}{x^4} = \frac{257}{16}\)
- Recall a mathematical identity: \((x^2 + \frac{1}{x^2})^2 = x^4 + \frac{1}{x^4} + 2\). Therefore, we can express \(x^4 + \frac{1}{x^4}\) in terms of \(x^2 + \frac{1}{x^2}\):
\((x^2 + \frac{1}{x^2})^2 = x^4 + \frac{1}{x^4} + 2\)
- Substitute the value of \(x^4 + \frac{1}{x^4}\):
\((x^2 + \frac{1}{x^2})^2 = \frac{257}{16} + 2 = \frac{289}{16}\)
- Taking the square root on both sides, we find:
\(x^2 + \frac{1}{x^2} = \frac{17}{4}\)
- Again, recall another identity: \((x + \frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2\). Use this to find \(x + \frac{1}{x}\):
\((x + \frac{1}{x})^2 = \frac{17}{4} + 2 = \frac{25}{4}\)
- Take the square root to get:
\(x + \frac{1}{x} = \frac{5}{2}\)
- Now, use the expression: \((x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x})\) to solve for \(x^3 + \frac{1}{x^3}\):
\((\frac{5}{2})^3 = x^3 + \frac{1}{x^3} + 3(\frac{5}{2})\)
- Calculate \((\frac{5}{2})^3\) and \(3(\frac{5}{2})\):
\(\frac{125}{8} = x^3 + \frac{1}{x^3} + \frac{15}{2}\)
- Convert \(\frac{15}{2}\) to a fraction with denominator 8:
\(\frac{15}{2} = \frac{60}{8}\)
- Substitute back to solve for \(x^3 + \frac{1}{x^3}\):
\(\frac{125}{8} = x^3 + \frac{1}{x^3} + \frac{60}{8}\)
- Rearrange the terms to find:
\(x^3 + \frac{1}{x^3} = \frac{125}{8} - \frac{60}{8} = \frac{65}{8}\)
Thus, the value of \(x^3 + \frac{1}{x^3}\) is \(\frac{65}{8}\), which corresponds to the correct answer given.
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