To determine the value of [tex]\(\sqrt{x - \sqrt{x}}\)[/tex] given the equation [tex]\(x^2 - 65x = 64\sqrt{x}\)[/tex], we will follow these steps:
1. Rewrite the equation in a friendly form for solving:
[tex]\[
x^2 - 65x - 64\sqrt{x} = 0
\][/tex]
2. Solve the quadratic equation in terms of [tex]\(x\)[/tex]:
[tex]\[
x = 0 \quad \text{or} \quad x = \frac{\sqrt{257}}{2} + \frac{129}{2}
\][/tex]
After solving, we get two solutions:
[tex]\[
x = 0
\][/tex]
and
[tex]\[
x = \frac{\sqrt{257}}{2} + \frac{129}{2}
\][/tex]
3. Consider the real solutions:
- For [tex]\(x = 0\)[/tex]:
[tex]\[
\sqrt{0 - \sqrt{0}} = \sqrt{0} = 0
\][/tex]
Therefore, the value of [tex]\(\sqrt{x - \sqrt{x}}\)[/tex] when [tex]\(x = 0\)[/tex] is [tex]\(0\)[/tex].
- For [tex]\(x = \frac{\sqrt{257}}{2} + \frac{129}{2}\)[/tex]:
[tex]\[
\sqrt{\left(\frac{\sqrt{257}}{2} + \frac{129}{2}\right) - \sqrt{\left(\frac{\sqrt{257}}{2} + \frac{129}{2}\right)}}
\][/tex]
Upon evaluating the above expression, it simplifies to a complicated algebraic expression that is not one of the provided options:
[tex]\[
\sqrt{-\sqrt{\left(\frac{\sqrt{257}}{2} + \frac{129}{2}\right)} + \left(\frac{\sqrt{257}}{2} + \frac{129}{2}\right)}
\][/tex]
From these steps and evaluating the real solutions, we find the primary candidate for the answer is when [tex]\(x = 0\)[/tex], which results in [tex]\(\sqrt{x - \sqrt{x}} = 0\)[/tex].
Thus, none of the provided options exactly match, but the closest discernible correct option given the allowable steps and constraints would be most logically formatted as:
[tex]\[
\boxed{0}
\][/tex]
