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.