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The expression you provided is:

\[
\sqrt{9x^{13}}
\]

We need to simplify this under the assumption that \( x \) is a positive real number.

### Step-by-step simplification:

1. **Factor the expression inside the square root:**
   \[
   \sqrt{9x^{13}} = \sqrt{9} \cdot \sqrt{x^{13}}
   \]

2. **Simplify \( \sqrt{9} \):**
   \[
   \sqrt{9} = 3
   \]

3. **Simplify \( \sqrt{x^{13}} \):**
   Use the property of square roots: \( \sqrt{x^a} = x^{a/2} \). So,
   \[
   \sqrt{x^{13}} = x^{13/2}
   \]

4. **Combine the results:**
   \[
   3 \cdot x^{13/2}
   \]

Thus, the simplified expression is:

\[
3x^{13/2}
\]

Would you like more details, or do you have any questions on how this was done?

### Relative questions:
1. What is the rule for simplifying square roots of powers, and why does \( x^{13} \) become \( x^{13/2} \)?
2. Can you explain why \( \sqrt{9} \) equals 3?
3. How would the answer change if \( x \) were a negative number?
4. How can this simplification be used in solving equations or practical applications?
5. What is the difference between simplifying expressions and solving equations?

### Tip:
When simplifying square roots involving exponents, always remember to halve the exponent of any variable under the radical.

Simplify \( \sqrt{9x^{13}} \) assuming that the variable represents a positive real number.

Learn how to simplify the square root of the expression \( \sqrt{9x^{13}} \) assuming that x is a positive real number. This problem covers key concepts like simplifying square roots and applying exponent rules, suitable for high school students in Grades 9-12.

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