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Let's analyze the given mathematical expression:

The expression is:

\[
\frac{(11)^{n+2} \times (121)^n - (121)^{\frac{3n}{2}}}{\left[(11)^{n+1} - (121)^{\frac{n}{2}}\right] (121)^n}
\]

### Solution Steps:

1. **Simplify Powers of 121:**
   Since \(121 = 11^2\), we can rewrite \(121^n\) and \(121^{\frac{3n}{2}}\) as powers of 11.
   - \(121^n = (11^2)^n = 11^{2n}\)
   - \(121^{\frac{3n}{2}} = (11^2)^{\frac{3n}{2}} = 11^{3n}\)
   - \(121^{\frac{n}{2}} = (11^2)^{\frac{n}{2}} = 11^n\)

   Substituting these into the original expression gives:

   \[
   \frac{(11)^{n+2} \times 11^{2n} - 11^{3n}}{\left[(11)^{n+1} - 11^n\right] 11^{2n}}
   \]

2. **Simplify the Numerator:**
   - The numerator becomes: 
     \[
     11^{n+2} \times 11^{2n} - 11^{3n} = 11^{n+2+2n} - 11^{3n} = 11^{3n+2} - 11^{3n}
     \]
     Factor out \(11^{3n}\) from the terms:
     \[
     11^{3n}(11^2 - 1) = 11^{3n}(121 - 1) = 11^{3n} \times 120
     \]

3. **Simplify the Denominator:**
   - The denominator is:
     \[
     \left[11^{n+1} - 11^n\right] 11^{2n}
     \]
     Factor out \(11^n\) from the terms inside the bracket:
     \[
     11^n(11 - 1) = 11^n \times 10
     \]
     So the denominator becomes:
     \[
     11^n \times 10 \times 11^{2n} = 11^{3n} \times 10
     \]

4. **Simplify the Overall Expression:**
   The expression now becomes:
   \[
   \frac{11^{3n} \times 120}{11^{3n} \times 10}
   \]
   Cancel out \(11^{3n}\) from both the numerator and the denominator:
   \[
   \frac{120}{10} = 12
   \]

Thus, the value of the given expression simplifies to **12**.

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Would you like further details or have any questions? Here are 5 related questions to expand on this:

1. How do you simplify expressions with powers and exponents?
2. What are the rules for factoring powers out of terms?
3. How can recognizing common bases help simplify complex expressions?
4. What other methods can be used to simplify rational expressions?
5. What is the general approach to solving algebraic fractions?

**Tip:** When dealing with powers, always try to express everything in terms of the same base for easier simplification.

Simplify the following expression:

\[
\frac{(11)^{n+2} \times (121)^n - (121)^{\frac{3n}{2}}}{\left[(11)^{n+1} - (121)^{\frac{n}{2}}\right] (121)^n}
\]

Learn how to simplify complex expressions with exponents by factoring powers of 11 and 121. This step-by-step solution explains how to use properties of exponents to simplify the given mathematical expression.

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