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The expression to simplify is:

\[
\left[\left(3x^3\right)^3\right]^3
\]

### Step 1: Simplify the innermost exponentiation.
First, simplify the expression inside the brackets:

\[
(3x^3)^3
\]

When you raise a product to a power, you raise each factor to that power:

\[
3^3 \cdot (x^3)^3
\]

This simplifies to:

\[
27x^{9}
\]

### Step 2: Apply the outer exponentiation.
Now, raise the entire expression \(27x^{9}\) to the power of 3:

\[
(27x^9)^3
\]

Again, apply the exponent to both the coefficient and the variable:

\[
27^3 \cdot (x^9)^3
\]

Simplify:

- \(27^3 = 19683\)
- \( (x^9)^3 = x^{27} \)

So, the expression simplifies to:

\[
19683x^{27}
\]

### Final Answer:
\[
\boxed{19683x^{27}}
\]

Would you like more details or have any questions? Here are some related questions to explore further:

1. How would the result change if the innermost expression were \((2x^2)^2\)?
2. Can you generalize the process to simplify \(\left[\left(ax^b\right)^c\right]^d\)?
3. What is the value of \((3^3)^3\) when expressed as a single power of 3?
4. How does the simplification process change if \(x\) has a coefficient with a different base?
5. How do you simplify the expression if it was \(\left[(2x^4)^2\right]^4\) instead?

**Tip:** Remember that when you raise a power to another power, you multiply the exponents.

Learn how to simplify the algebraic expression (3x^3)^9 step-by-step. This problem involves applying the power rule and exponentiation principles to simplify nested expressions. Suitable for high school students and anyone preparing for algebraic simplification challenges.

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