Let's solve the expression step by step:

\[
-4\left(\frac{1}{3} \cdot \sqrt{ \left( 5 \frac{1}{2} \right) \left( 5 \frac{2}{3} \right) }\right)
\]

### Step 1: Convert mixed fractions to improper fractions
\[
5 \frac{1}{2} = \frac{11}{2}, \quad 5 \frac{2}{3} = \frac{17}{3}
\]

### Step 2: Multiply the two fractions
\[
\left( \frac{11}{2} \right) \left( \frac{17}{3} \right) = \frac{11 \times 17}{2 \times 3} = \frac{187}{6}
\]

### Step 3: Take the square root
\[
\sqrt{\frac{187}{6}} = \frac{\sqrt{187}}{\sqrt{6}}
\]

### Step 4: Multiply by \(\frac{1}{3}\)
\[
\frac{1}{3} \cdot \frac{\sqrt{187}}{\sqrt{6}} = \frac{\sqrt{187}}{3\sqrt{6}}
\]

### Step 5: Multiply by \(-4\)
\[
-4 \cdot \frac{\sqrt{187}}{3\sqrt{6}} = -\frac{4\sqrt{187}}{3\sqrt{6}}
\]

### Final Simplification (optional)
You can leave the expression as:

\[
-\frac{4\sqrt{187}}{3\sqrt{6}}
\]

Or, rationalize the denominator:

\[
-\frac{4\sqrt{187}}{3\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = -\frac{4\sqrt{1122}}{18}
\]

Which is the simplified result.

### Result:
\[
-\frac{2\sqrt{1122}}{9}
\]

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Do you want more details on any step, or do you have further questions?

Here are some related questions:
1. What is the process for converting mixed numbers to improper fractions?
2. How do you multiply two fractions?
3. What is the significance of rationalizing the denominator?
4. How do you simplify square roots of large numbers?
5. What are some techniques to solve radical expressions faster?

**Tip**: When rationalizing denominators, always multiply by a form of 1 (like \(\frac{\sqrt{a}}{\sqrt{a}}\)) to maintain equivalence!

This page explains how to simplify the expression -4(⅓√[(5½)(5⅔)]) step by step. Learn to convert mixed fractions to improper fractions, multiply fractions, handle square roots, and rationalize denominators. Perfect for students in Grades 9-12, covering essential algebraic and fraction concepts.

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