To solve \( \sqrt{\sqrt{256x^4}} \), we'll break it down step by step:

### Step 1: Simplify the inner square root
We first focus on simplifying \( \sqrt{256x^4} \).

\[
\sqrt{256x^4} = \sqrt{256} \times \sqrt{x^4}
\]

We know that:
\[
\sqrt{256} = 16
\quad \text{and} \quad \sqrt{x^4} = x^2
\]

Thus, we get:
\[
\sqrt{256x^4} = 16x^2
\]

### Step 2: Apply the outer square root
Now, we need to simplify \( \sqrt{16x^2} \).

\[
\sqrt{16x^2} = \sqrt{16} \times \sqrt{x^2}
\]

Since:
\[
\sqrt{16} = 4
\quad \text{and} \quad \sqrt{x^2} = |x|
\]

Thus, the final result is:
\[
\sqrt{\sqrt{256x^4}} = 4|x|
\]

The absolute value is needed because \( \sqrt{x^2} \) gives the positive root.

### Final Answer:
\[
\boxed{4|x|}
\]

Would you like more details on this process, or do you have any questions?

Here are some related questions you might find interesting:
1. How do you simplify \( \sqrt{x^6} \)?
2. What is the general rule for simplifying nested square roots?
3. Can you simplify \( \sqrt{81x^8} \)?
4. How does the absolute value affect the simplification of square roots?
5. What happens if \( x \) is negative in the expression \( \sqrt{x^2} \)?

**Tip:** Always remember that the square root of a variable squared, \( \sqrt{x^2} \), equals \( |x| \), not just \( x \). This accounts for both positive and negative values of \( x \).

Explore how to simplify the nested square root expression \( \sqrt{\sqrt{256x^4}} \). This solution involves understanding exponents, square roots, and absolute values. Ideal for students in Grades 9-11 learning about algebraic simplifications.

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