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The given expression appears to be a complex fraction with mixed numbers and negative exponents. Let's break it down and solve it step-by-step:

The expression is:

\[
\frac{\left( 5 \frac{3}{5} - 8.8 \right) \cdot \frac{5}{6} \cdot \left( -\frac{7}{3} \right)^{-1}}{\left( 4 \frac{2}{3} - 4 \frac{1}{6} \right) \cdot \left( \left( -2 \right)^{-1} \right)^{-3}}
\]

### Step 1: Convert mixed numbers into improper fractions
- \(5 \frac{3}{5} = \frac{28}{5}\)
- \(4 \frac{2}{3} = \frac{14}{3}\)
- \(4 \frac{1}{6} = \frac{25}{6}\)

### Step 2: Simplify the fractions
The given expression becomes:

\[
\frac{\left( \frac{28}{5} - 8.8 \right) \cdot \frac{5}{6} \cdot \left( -\frac{7}{3} \right)^{-1}}{\left( \frac{14}{3} - \frac{25}{6} \right) \cdot \left( \left( -2 \right)^{-1} \right)^{-3}}
\]

Now, let's simplify the terms inside the parentheses.

#### Numerator simplification:
\[
\frac{28}{5} - 8.8 = \frac{28}{5} - \frac{88}{10} = \frac{28}{5} - \frac{44}{5} = \frac{-16}{5}
\]
\[
\left( -\frac{7}{3} \right)^{-1} = -\frac{3}{7}
\]

So, the numerator becomes:
\[
\frac{-16}{5} \cdot \frac{5}{6} \cdot -\frac{3}{7}
\]

#### Denominator simplification:
\[
\frac{14}{3} - \frac{25}{6} = \frac{28}{6} - \frac{25}{6} = \frac{3}{6} = \frac{1}{2}
\]
\[
\left( \left( -2 \right)^{-1} \right)^{-3} = \left( -\frac{1}{2} \right)^{-3} = -8
\]

So, the denominator becomes:
\[
\frac{1}{2} \cdot (-8)
\]

### Step 3: Final simplification

Numerator:
\[
\frac{-16}{5} \cdot \frac{5}{6} \cdot -\frac{3}{7} = \frac{48}{42} = \frac{24}{21} = \frac{8}{7}
\]

Denominator:
\[
\frac{1}{2} \cdot (-8) = -4
\]

So, the entire expression becomes:
\[
\frac{\frac{8}{7}}{-4} = \frac{8}{7} \cdot \frac{1}{-4} = -\frac{2}{7}
\]

### Final Answer:
\[
- \frac{2}{7}
\]

Let me know if you want more details or have any questions!

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Here are five related questions for further practice:
1. How would you handle an expression with mixed fractions and exponents of zero?
2. What is the effect of a negative exponent on a fraction?
3. Can you simplify expressions involving powers of negative numbers?
4. How would you simplify a complex fraction with both addition and subtraction?
5. How do you convert between improper fractions and mixed numbers?

**Tip:** Always ensure fractions are simplified as much as possible before performing arithmetic operations, especially when exponents are involved.

Simplify the following complex fraction: 

( (5 3/5 - 8.8) * 5/6 * (-7/3)^-1 ) / ( (4 2/3 - 4 1/6) * ( (-2)^-1 )^-3 ).

Learn how to simplify a complex fraction involving mixed numbers, fractions, and negative exponents. This problem includes converting mixed numbers to improper fractions, applying exponentiation rules, and performing arithmetic operations on fractions. Suitable for students in Grades 9-10.

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