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The mathematical problem asks to simplify the following expression using the least common multiple (LCM) and reduce it to its simplest form. Here's the expression provided:

\[
\left( \frac{\frac{11}{5} - \frac{2}{31}}{\frac{3}{21}} \right)^3 \times \frac{11}{9}
\]

### Step 1: Simplify the expression inside the parentheses

#### Step 1a: Find a common denominator for the terms in the numerator \(\frac{11}{5} - \frac{2}{31}\).

The least common multiple of 5 and 31 is 155, so we rewrite the fractions:
\[
\frac{11}{5} = \frac{11 \times 31}{5 \times 31} = \frac{341}{155}, \quad \frac{2}{31} = \frac{2 \times 5}{31 \times 5} = \frac{10}{155}
\]
Now subtract the fractions:
\[
\frac{341}{155} - \frac{10}{155} = \frac{331}{155}
\]

#### Step 1b: Divide by \(\frac{3}{21}\).
\[
\frac{\frac{331}{155}}{\frac{3}{21}} = \frac{331}{155} \times \frac{21}{3} = \frac{331 \times 21}{155 \times 3} = \frac{6951}{465}
\]
Simplify the fraction:
\[
\frac{6951}{465} = \frac{6951 \div 3}{465 \div 3} = \frac{2317}{155}
\]

### Step 2: Apply the exponent \(3\)
Now, cube the simplified fraction:
\[
\left( \frac{2317}{155} \right)^3 = \frac{2317^3}{155^3} = \frac{1242887233}{3723875}
\]

### Step 3: Multiply by \(\frac{11}{9}\)
Now multiply the result by \(\frac{11}{9}\):
\[
\frac{1242887233}{3723875} \times \frac{11}{9} = \frac{1242887233 \times 11}{3723875 \times 9} = \frac{13671759563}{33514875}
\]

### Final Answer:
The simplified expression is:
\[
\frac{13671759563}{33514875}
\]
This fraction can be simplified further if common factors exist, but in this case, it appears to be in its simplest form.

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the process for finding the least common multiple (LCM) of two numbers?
2. How can you simplify complex fractions involving multiple operations?
3. What are some common methods for factoring large numbers to simplify fractions?
4. How do you handle exponents when they apply to fractions?
5. How can multiplying fractions help simplify complex expressions?

**Tip:** When dealing with complex fractions, always break the problem down step by step, simplifying small parts before combining them.

Simplify the expression using LCM and reduce it to its simplest form: (11/5 - 2/31) / (3/21) * (11/9)

Learn how to simplify a complex fraction by finding the least common multiple (LCM) of the denominators and reducing the expression to its simplest form. This problem is ideal for students in Grades 9-10, focusing on algebraic fractions and simplification techniques.

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