The given expression seems unclear, but I will assume you're attempting to write something like:

\[
\sqrt{\log \left( \frac{4x^2 - x^2}{3} \right)}
\]

### Step-by-Step Solution:

1. **Simplify the fraction:**

\[
\frac{4x^2 - x^2}{3} = \frac{3x^2}{3} = x^2
\]

2. **Apply the logarithm:**

\[
\log(x^2)
\]

Using the logarithmic identity \(\log(a^b) = b\log(a)\), we simplify it further:

\[
\log(x^2) = 2\log(x)
\]

3. **Apply the square root:**

\[
\sqrt{2\log(x)}
\]

Thus, the simplified form of the expression is:

\[
\sqrt{2\log(x)}
\]

### Final Answer:
\[
\boxed{\sqrt{2\log(x)}}
\]

Would you like more details or have any questions about this? 

Here are 5 related questions to explore:
1. What are the properties of logarithms that can simplify expressions?
2. How does the square root function interact with logarithmic expressions?
3. Can logarithms be applied to negative numbers or zero?
4. What happens when you take the derivative of \(\sqrt{2\log(x)}\)?
5. How would the expression change if the base of the logarithm is not 10 (e.g., natural log)?

**Tip:** Always check for domain restrictions when dealing with logarithms, since \(\log(x)\) is only defined for \(x > 0\).

The query submitted by the user: \sqrt{log \frac{4x2-x2|3}

This problem involves simplifying the expression \sqrt{log \frac{4x^2 - x^2}{3}}. Learn how to apply logarithmic identities and the square root function to arrive at the solution \sqrt{2log(x)}. Suitable for students in Grades 9-12, covering logarithmic and algebraic concepts.

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