Let's break down and simplify the expression step by step:

Given expression: 

\[
9^2 \cdot 3^5 \cdot \frac{1}{27} \cdot \left( \frac{1}{3} \right)^{-4}
\]

### Step 1: Simplify each term

1. **Simplify \(9^2\)**:  
   \[
   9^2 = 81
   \]
   
2. **Simplify \(3^5\)**:  
   \[
   3^5 = 243
   \]

3. **Simplify \(\frac{1}{27}\)**:  
   \[
   \frac{1}{27} = 3^{-3}
   \]

4. **Simplify \(\left( \frac{1}{3} \right)^{-4}\)**:  
   Using the property \(\left( \frac{1}{a} \right)^{-b} = a^b\):
   \[
   \left( \frac{1}{3} \right)^{-4} = 3^4
   \]

### Step 2: Combine powers of 3
Now let's combine the powers of 3.

- We have:
  \[
  81 \cdot 243 \cdot 3^{-3} \cdot 3^4
  \]

- Combine the exponents of 3:
  \[
  3^5 \cdot 3^{-3} \cdot 3^4 = 3^{5 - 3 + 4} = 3^6
  \]

### Step 3: Final multiplication

Now, simplify the entire expression:

\[
81 \cdot 243 \cdot 3^6
\]

We know that \(81 = 3^4\) and \(243 = 3^5\), so the entire expression is:

\[
3^4 \cdot 3^5 \cdot 3^6 = 3^{4 + 5 + 6} = 3^{15}
\]

Thus, the simplified result of the expression is:

\[
3^{15}
\]

If you want this in numerical form:

\[
3^{15} = 14,348,907
\]

---

Would you like more details or have any questions?

### Related questions:
1. How do you simplify expressions involving negative exponents?
2. What are the rules for multiplying powers with the same base?
3. How can you express \(9^2\) in terms of powers of 3?
4. How do fractional exponents work in similar problems?
5. Can this approach be used for other types of powers (e.g., with bases other than 3)?

### Tip:
When working with exponents, always try to express terms with a common base to simplify your work.

Simplify the expression 9^2 * 3^5 * 1/27 * (1/3)^-4 with step-by-step explanations. Learn about exponents, negative exponents, and how to multiply powers with the same base. Suitable for high school students (Grades 9-12).

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