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The given expression for \( P \) appears to involve multiple nested cube roots. Let’s break it down step by step.

The expression is:

\[
P = \left( \sqrt[3]{\sqrt[3]{\sqrt[3]{\sqrt[3]{3 \cdot \sqrt[3]{3}}}}} \right)^3
\]

We can simplify this by removing the cube and cube roots step by step:

1. First, simplify the outermost cube root:
   \[
   P = \left( \sqrt[3]{\sqrt[3]{\sqrt[3]{3 \cdot \sqrt[3]{3}}}} \right)^3
   \]
   The cube and cube root cancel out, giving:
   \[
   P = \sqrt[3]{\sqrt[3]{\sqrt[3]{3 \cdot \sqrt[3]{3}}}}
   \]

2. Now, simplify the remaining cube roots layer by layer:
   - Let’s start with \( 3 \cdot \sqrt[3]{3} \). This simplifies as:
     \[
     3 \cdot \sqrt[3]{3} = 3^{1 + \frac{1}{3}} = 3^{\frac{4}{3}}
     \]
   - Now apply the cube root:
     \[
     \sqrt[3]{3^{\frac{4}{3}}} = 3^{\frac{4}{9}}
     \]

3. Apply the next cube root:
   \[
   \sqrt[3]{3^{\frac{4}{9}}} = 3^{\frac{4}{27}}
   \]

4. Apply the final cube root:
   \[
   \sqrt[3]{3^{\frac{4}{27}}} = 3^{\frac{4}{81}}
   \]

Thus, we find:

\[
P = 3^{\frac{4}{81}}
\]

This is the simplified form of the expression.

Let me know if you'd like any more details or further clarification!

### Related Questions:
1. What is the significance of cube roots in real-world applications?
2. How would the process differ if we used square roots instead of cube roots?
3. What are some examples of nested roots in other mathematical problems?
4. How can we generalize this problem for higher exponents or roots?
5. How does simplifying nested radicals affect the precision of the result?

### Tip:
When simplifying nested radicals, it's helpful to keep track of the powers step-by-step to avoid mistakes with the exponents.

Simplify the expression: P = ((3^(1/3 * 1/3 * 1/3))^3)

Learn how to simplify a nested cube roots expression step by step using the laws of exponents. This problem involves exponents and radicals, suitable for Grades 9-12 math students.

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