To solve for $X$ in the given equation, let’s analyze each part carefully.

The equation reads:

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

Step 1: Simplify the Left Side

1. Calculate $25^0$: Any number raised to the power of 0 is 1. So, $25^0 = 1$.

2. Raise to the Power of 3: Since $(25^0)^3 = 1^3 = 1$.

3. Add $\frac{3}{4}$: $1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}$

4. Divide by $\frac{7}{256X}$: $\frac{\frac{7}{4}}{\frac{7}{256X}} = \frac{7}{4} \times \frac{256X}{7} = \frac{256X}{4} = 64X$

Thus, the left side simplifies to $\sqrt{64X} = 8\sqrt{X}$.

Step 2: Simplify the Right Side

1. Simplify $4X^2 + 256$: This term remains as it is in the numerator.

2. Simplify the Denominator:

   • First, calculate $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
   • Thus, $\frac{120}{5!} = \frac{120}{120} = 1$.

   Since $1^{256} = 1$, the denominator simplifies to 1.

So, the right side simplifies to $\sqrt{4X^2 + 256}$.

Step 3: Set Up the Simplified Equation

After simplifying, we have: $8\sqrt{X} = \sqrt{4X^2 + 256}$

Step 4: Square Both Sides

To eliminate the square roots, square both sides: $(8\sqrt{X})^2 = (\sqrt{4X^2 + 256})^2$ $64X = 4X^2 + 256$

Step 5: Rearrange and Solve for $X$

Bring all terms to one side to form a quadratic equation: $4X^2 - 64X + 256 = 0$

Divide by 4 to simplify: $X^2 - 16X + 64 = 0$

This is a quadratic equation, and it can be factored as: $(X - 8)^2 = 0$

So, $X = 8$.

Final Answer

$X = 8$

Would you like further details on any step, or have any questions?

Here are some related questions for practice:

1. What is the result of simplifying expressions with zero exponents, like $25^0$?
2. How does the square root operation affect equations when both sides are squared?
3. Why is it important to check for extraneous solutions when solving radical equations?
4. What are the general steps for solving a quadratic equation?
5. How does simplifying factorials help in calculations?

Tip: Always simplify each component separately to avoid errors when dealing with complex expressions.