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The expression given is:

\[
\frac{x^3 \cdot \sqrt[3]{x^3} \cdot x^5}{x^{-1/3}}
\]

We need to simplify this into the form \( x^n \) and determine the value of \( n \).

### Step-by-step solution:

1. **Cube Root of \( x^3 \)**:
   \[
   \sqrt[3]{x^3} = x^{3/3} = x^1
   \]

2. **Multiply the terms in the numerator**:
   \[
   x^3 \cdot x^1 \cdot x^5 = x^{3+1+5} = x^9
   \]

3. **Simplify the denominator**:
   \[
   x^{-1/3}
   \]

4. **Combine the numerator and denominator** using the rule \( \frac{x^a}{x^b} = x^{a-b} \):
   \[
   \frac{x^9}{x^{-1/3}} = x^{9 - (-1/3)} = x^{9 + 1/3}
   \]

5. **Final expression**:
   \[
   x^{9 + 1/3} = x^{28/3}
   \]

Thus, the value of \( n \) is:

\[
n = \frac{28}{3}
\]

Would you like further details or have any questions?

Here are some related questions:
1. What are the rules for simplifying exponents with the same base?
2. How do you simplify expressions involving cube roots?
3. What is the general rule for dividing exponents with the same base?
4. How would this expression change if it involved a square root instead of a cube root?
5. Can this approach be applied to more complex exponential expressions?

**Tip**: Always remember to convert roots into fractional exponents to simplify exponent expressions efficiently!

The expression \(\frac{x^3 \cdot \sqrt[3]{x^3} \cdot x^5}{x^{-1/3}}\) is written as a single exponential \(x^n\). What is the value of \(n\)?

This problem involves simplifying the expression \(\frac{x^3 \cdot \sqrt[3]{x^3} \cdot x^5}{x^{-1/3}}\) into the form \(x^n\). It covers concepts like exponents, roots, and the rules for multiplying and dividing exponents. Suitable for students in Grades 10-12.

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