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!

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.