https://file.aiquickdraw.com/mathbot/file/1729102030298zkmxjr70.jpg

The expression you provided is:

\[
\left( \sqrt{\left( 2\sqrt{3} \right)^4} \right)^2
\]

Let's solve this step by step:

### Step 1: Simplify the innermost expression \((2\sqrt{3})^4\)

\[
(2\sqrt{3})^4 = (2^4) \cdot (\sqrt{3})^4
\]

- \( 2^4 = 16 \)
- \((\sqrt{3})^4 = (\sqrt{3})^2 \cdot (\sqrt{3})^2 = 3 \cdot 3 = 9\)

Thus, 

\[
(2\sqrt{3})^4 = 16 \cdot 9 = 144
\]

### Step 2: Apply the square root \(\sqrt{144}\)

The square root of 144 is 12:

\[
\sqrt{144} = 12
\]

### Step 3: Apply the outer exponent \((12)^2\)

Now square 12:

\[
12^2 = 144
\]

### Final Answer:
The value of the given expression is \( \boxed{144} \).

Let me know if you'd like a more detailed explanation or have any questions! Here are five related questions to expand your understanding:

1. How do you handle exponents of square roots in general?
2. What are the properties of exponents that were used in this problem?
3. Can you explain why \(\sqrt{144} = 12\)?
4. What is the significance of powers and square roots in simplifying radical expressions?
5. What happens if the base inside the root were a negative number?

**Tip:** When dealing with nested radicals and exponents, always simplify the innermost part first and work your way outwards for clarity.

Simplify the expression: ( ( sqrt( ( 2sqrt(3) )^4 ) )^2 ).

Learn how to simplify the expression ( ( sqrt( ( 2sqrt(3) )^4 ) )^2 ) with a detailed, step-by-step solution. Explore key concepts such as exponents, square roots, and their properties. This problem is ideal for students in Grades 8-10.

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