Let's analyze the given equation and solve for $X$.

The equation is:

$\sqrt{\frac{(25^0)^3 + \frac{3}{4}}{\frac{7}{256x}}} = \sqrt{\frac{4x^2 + 256}{\frac{120}{5!}}^{256}}$

Step-by-step analysis:

1. Simplify both sides:

   • Left side:

     • $25^0 = 1$, so the numerator becomes $1^3 + \frac{3}{4} = 1 + \frac{3}{4} = \frac{7}{4}$.
     • Therefore, the left side is: $\sqrt{\frac{\frac{7}{4}}{\frac{7}{256x}}} = \sqrt{\frac{7}{4} \times \frac{256x}{7}} = \sqrt{\frac{256x}{4}} = \sqrt{64x}.$

   • Right side:

     • $5! = 120$, so $\frac{120}{5!} = \frac{120}{120} = 1$, and hence $\left( \frac{120}{5!} \right)^{256} = 1^{256} = 1$.
     • Therefore, the right side becomes: $\sqrt{\frac{4x^2 + 256}{1}} = \sqrt{4x^2 + 256}.$

2. Equate both sides: $\sqrt{64x} = \sqrt{4x^2 + 256}.$

3. Square both sides to eliminate the square roots: $64x = 4x^2 + 256.$

4. Rearrange the equation: $4x^2 - 64x + 256 = 0.$

5. Divide the entire equation by 4 to simplify: $x^2 - 16x + 64 = 0.$

6. Solve the quadratic equation using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -16$, and $c = 64$: $x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(64)}}{2(1)} = \frac{16 \pm \sqrt{256 - 256}}{2} = \frac{16 \pm 0}{2}.$ $x = \frac{16}{2} = 8.$

Thus, the value of $X$ is $\boxed{8}$.

Would you like a further breakdown or additional details?

Here are 5 related questions for practice:

1. How do you apply the quadratic formula to solve any quadratic equation?
2. What is the general approach to simplifying nested square root expressions?
3. What are the steps involved in factoring quadratic equations?
4. How do factorials and powers impact complex equations like this one?
5. What techniques can help verify the solutions to equations?

Tip: Always check the domain of your solutions when working with square roots to ensure the values of the variable are valid.