Let's break down the given expression step by step:

$\left[ \frac{3^7}{(3^2 \times 3)^3} \right]^{-2}$

Step 1: Simplify the denominator

First, simplify the expression inside the parentheses in the denominator:

$3^2 \times 3 = 3^{2+1} = 3^3$

So the denominator becomes:

$(3^3)^3 = 3^{3 \times 3} = 3^9$

Step 2: Substitute back into the original expression

Now, the expression becomes:

$\left[ \frac{3^7}{3^9} \right]^{-2}$

Step 3: Simplify the fraction

Use the property of exponents, where $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{3^7}{3^9} = 3^{7-9} = 3^{-2}$

Now, the expression is:

$\left( 3^{-2} \right)^{-2}$

Step 4: Apply the power of a power rule

Use the rule $(a^m)^n = a^{m \times n}$:

$\left( 3^{-2} \right)^{-2} = 3^{-2 \times -2} = 3^4$

Final Answer:

$3^4 = 81$

So, the value of the expression is:

$\boxed{81}$

Would you like more details on any part of this process? Or do you have any further questions?

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Here are some related questions to help you explore:

1. How do you simplify powers of powers in exponential expressions?
2. What happens when you divide terms with the same base but different exponents?
3. How do you simplify expressions involving negative exponents?
4. Can you explain the properties of exponents in more detail?
5. What are the real-world applications of exponentiation?

Tip: When simplifying expressions with exponents, always apply the exponent rules systematically to avoid errors—especially when dealing with negative exponents!