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:

1. Firstly, solve the given equation for \(x^4 + \frac{1}{x^4}\):

\(16(x^4 + \frac{1}{x^4}) = 257\)

1. Divide both sides by 16 to isolate \(x^4 + \frac{1}{x^4}\):

\(x^4 + \frac{1}{x^4} = \frac{257}{16}\)

1. 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 \(x^2 + \frac{1}{x^2}\):

\((x^2 + \frac{1}{x^2})^2 = x^4 + \frac{1}{x^4} + 2\)

1. Substitute the value of \(x^4 + \frac{1}{x^4}\):

\((x^2 + \frac{1}{x^2})^2 = \frac{257}{16} + 2 = \frac{289}{16}\)

1. Taking the square root on both sides, we find:

\(x^2 + \frac{1}{x^2} = \frac{17}{4}\)

1. Again, recall another identity: \((x + \frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2\). Use this to find \(x + \frac{1}{x}\):

\((x + \frac{1}{x})^2 = \frac{17}{4} + 2 = \frac{25}{4}\)

1. Take the square root to get:

\(x + \frac{1}{x} = \frac{5}{2}\)

1. Now, use the expression: \((x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x})\) to solve for \(x^3 + \frac{1}{x^3}\):

\((\frac{5}{2})^3 = x^3 + \frac{1}{x^3} + 3(\frac{5}{2})\)

1. Calculate \((\frac{5}{2})^3\) and \(3(\frac{5}{2})\):

\(\frac{125}{8} = x^3 + \frac{1}{x^3} + \frac{15}{2}\)

1. Convert \(\frac{15}{2}\) to a fraction with denominator 8:

\(\frac{15}{2} = \frac{60}{8}\)

1. Substitute back to solve for \(x^3 + \frac{1}{x^3}\):

\(\frac{125}{8} = x^3 + \frac{1}{x^3} + \frac{60}{8}\)

1. 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.