https://file.aiquickdraw.com/mathbot/file/1727643779669xzwkvqzr.jpg

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**.

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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.

Simplify the following radical expression: \(\sqrt{\frac{\sqrt[3]{\sqrt{64} + 4\sqrt{256}}}{\sqrt{64} + \sqrt{256}}}\)

This problem simplifies a complex radical expression involving cube roots and square roots. Using key algebraic concepts, the solution simplifies \(\sqrt{\frac{\sqrt[3]{\sqrt{64} + 4\sqrt{256}}}{\sqrt{64} + \sqrt{256}}}\) step by step. Ideal for students in Grades 10-12.

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