[(3-2(1/2)4(3)(1/3)^1/2-(4(-2)/(1/3)]^1/2

Let's break down the expression step by step to simplify and then solve it:

Given expression:

\[
\left( 3 - 2 \times \frac{1}{2} \times 4 \times 3 \times \left(\frac{1}{3}\right)^{1/2} - \left(\frac{4 \times (-2)}{\frac{1}{3}}\right) \right)^{1/2}
\]

### Step 1: Simplify inside the square root

#### 1.1 Simplify \( 2 \times \frac{1}{2} \times 4 \times 3 \times \left(\frac{1}{3}\right)^{1/2} \)
- \( 2 \times \frac{1}{2} = 1 \)
- \( 1 \times 4 = 4 \)
- \( 4 \times 3 = 12 \)
- \( \left(\frac{1}{3}\right)^{1/2} = \frac{1}{\sqrt{3}} \)

Thus, \( 12 \times \frac{1}{\sqrt{3}} = \frac{12}{\sqrt{3}} \).

To rationalize the denominator:

\[
\frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12\sqrt{3}}{3} = 4\sqrt{3}
\]

#### 1.2 Simplify \( \left(\frac{4 \times (-2)}{\frac{1}{3}}\right) \)
- \( 4 \times (-2) = -8 \)
- Dividing by \(\frac{1}{3}\) is equivalent to multiplying by 3:
\[
\frac{-8}{\frac{1}{3}} = -8 \times 3 = -24
\]

### Step 2: Substitute the simplified values back into the expression:

\[
\left( 3 - 4\sqrt{3} - (-24) \right)^{1/2}
\]

Simplify the expression inside the square root:

\[
\left( 3 - 4\sqrt{3} + 24 \right)^{1/2}
\]

\[
\left( 27 - 4\sqrt{3} \right)^{1/2}
\]

### Final Answer:
The simplified expression is:

\[
\sqrt{27 - 4\sqrt{3}}
\]

This is the most simplified form of the given expression.

Would you like a detailed explanation of any specific steps? 

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Here are some related questions:

1. How do you simplify square root expressions further when irrational numbers are involved?
2. What are the rules for rationalizing denominators?
3. Can you explain the process of multiplying and dividing fractions in more detail?
4. How do you handle nested operations within an expression?
5. Why is it important to follow the order of operations in complex expressions?

**Tip:** Always simplify expressions step by step to avoid mistakes, especially when dealing with roots and fractions.

Learn how to simplify complex radical expressions involving fractions and nested operations with a step-by-step breakdown. This problem covers key algebraic concepts, including the order of operations, properties of radicals, and rationalization of denominators. Suitable for students in Grades 10-12.

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