The given expression in the image can be interpreted as:

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

Step-by-step breakdown:

1. Simplify individual square roots:

   • $\sqrt{64} = 8$
   • $\sqrt{256} = 16$

2. Numerator simplification:

   • Inside the cube root: $\sqrt[3]{8 + 4 \times 16}$
   • This becomes: $\sqrt[3]{8 + 64} = \sqrt[3]{72}$

3. Denominator simplification:

   • $\sqrt{64} + \sqrt{256} = 8 + 16 = 24$

4. The expression now becomes: $\sqrt{\frac{\sqrt[3]{72}}{24}}$

5. Approximate cube root:

   • $\sqrt[3]{72} \approx 4.16$

6. Now we have: $\sqrt{\frac{4.16}{24}} \approx \sqrt{0.1733} \approx 0.416$

Thus, the simplified result of the expression is approximately 0.416.

Would you like more details on any part of this solution, or do you have any questions?

Here are 5 related questions for practice:

1. How do you simplify cube roots of non-perfect cubes?
2. What is the relationship between cube roots and square roots in nested expressions?
3. Can you express $\sqrt{256}$ in terms of powers of 2?
4. How do you approximate cube roots without a calculator?
5. What is the general approach to simplify expressions with multiple nested radicals?

Tip: For non-perfect cubes, breaking down the number into prime factors can help approximate cube roots more accurately.