Let's simplify the given expression step by step:

[ \sqrt{\frac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64} + \sqrt{256}}} ]

Step 1: Simplify the roots individually.

1. $\sqrt[3]{64}$:

   • $64 = 4^3$, so $\sqrt[3]{64} = 4$.

2. $\sqrt[4]{256}$:

   • $256 = 4^4$, so $\sqrt[4]{256} = 4$.

3. $\sqrt{64}$:

   • $64 = 8^2$, so $\sqrt{64} = 8$.

4. $\sqrt{256}$:

   • $256 = 16^2$, so $\sqrt{256} = 16$.

Step 2: Substitute the simplified values back into the expression.

The expression becomes:

$\sqrt{\frac{4 + 4}{8 + 16}} = \sqrt{\frac{8}{24}} = \sqrt{\frac{1}{3}}.$

Step 3: Simplify the square root.

$\sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}}.$

Final Answer:

$\boxed{\frac{1}{\sqrt{3}}}.$

Would you like any further details or have any questions?

Here are some related questions to expand on this:

1. What are cube roots and how are they different from square roots?
2. How can you rationalize a denominator containing a square root?
3. What is the general process for simplifying expressions with multiple roots?
4. How do fractional exponents relate to roots like $\sqrt[3]{64}$?
5. How can simplifying roots help in solving complex algebraic equations?

Tip: To simplify expressions faster, always start by breaking down each root into its prime factors!