Let's solve the given expression step by step.

### Expression:
\[
\left(\frac{1 \times 1}{5} - \frac{2}{3 \times 1}\right)^3 \div \left(\frac{3}{2 \times 1} \times \frac{11}{9}\right)
\]

### Steps:

1. **Simplify the first term inside the parentheses:**
   \[
   \frac{1 \times 1}{5} = \frac{1}{5}
   \]
   
2. **Simplify the second term inside the parentheses:**
   \[
   \frac{2}{3 \times 1} = \frac{2}{3}
   \]
   
3. **Combine the two terms inside the parentheses:**
   \[
   \frac{1}{5} - \frac{2}{3}
   \]
   To subtract these fractions, find a common denominator:
   \[
   \frac{1}{5} = \frac{3}{15}, \quad \frac{2}{3} = \frac{10}{15}
   \]
   \[
   \frac{3}{15} - \frac{10}{15} = \frac{-7}{15}
   \]
   
4. **Cube the result:**
   \[
   \left(\frac{-7}{15}\right)^3 = \frac{-343}{3375}
   \]
   
5. **Simplify the denominator:**
   \[
   \frac{3}{2 \times 1} = \frac{3}{2}
   \]
   Multiply by \(\frac{11}{9}\):
   \[
   \frac{3}{2} \times \frac{11}{9} = \frac{33}{18} = \frac{11}{6}
   \]

6. **Final division:**
   \[
   \frac{\frac{-343}{3375}}{\frac{11}{6}} = \frac{-343}{3375} \times \frac{6}{11} = \frac{-2058}{37125} 
   \]

### Simplification:
Let's simplify the final fraction by finding the greatest common divisor (GCD):

\[
\frac{-2058}{37125} = \frac{-686}{12375}
\]

The GCD of 686 and 12375 is 1, so this fraction is already in its simplest form.

### Final Answer:
\[
\frac{-686}{12375}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you find the common denominator when adding or subtracting fractions?
2. What is the process for simplifying a fraction to its lowest terms?
3. How do you handle operations involving both fractions and negative numbers?
4. Can you explain how to cube a fraction and simplify the result?
5. What are the steps for dividing complex fractions?

**Tip:** When dealing with complex expressions involving fractions, always simplify each part as much as possible before combining them.

Learn how to solve the mathematical expression ((1*1)/5 - 2/(3*1))^3 / (3/(2*1) * 11/9) step-by-step with a detailed solution. This problem involves fraction arithmetic, exponentiation, and division of fractions. Suitable for high school students.

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