https://file.aiquickdraw.com/mathbot/file/1726332729294sh4qge4y.jpg

The expression shown in the image is:

\[
\left( \frac{3n^5}{2m^7} \right)^3
\]

We are asked to simplify it and express the answer using only positive exponents.

### Step-by-step solution:

1. **Apply the power of a quotient rule:**

   \[
   \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}
   \]

   Apply this rule to the expression:

   \[
   \left( \frac{3n^5}{2m^7} \right)^3 = \frac{(3n^5)^3}{(2m^7)^3}
   \]

2. **Simplify the numerator and denominator:**

   - Numerator: \( (3n^5)^3 = 3^3 \cdot (n^5)^3 = 27n^{15} \)
   - Denominator: \( (2m^7)^3 = 2^3 \cdot (m^7)^3 = 8m^{21} \)

   Therefore, the expression becomes:

   \[
   \frac{27n^{15}}{8m^{21}}
   \]

This is the simplified form of the expression with only positive exponents.

### Final Answer:
\[
\frac{27n^{15}}{8m^{21}}
\]

---

Do you need any further clarifications or details on any step?

Here are 5 related questions to explore the topic further:

1. How would you simplify an expression with negative exponents?
2. What is the power of a product rule in exponents?
3. How does the power of a power rule work with exponents?
4. Can you explain the process of simplifying fractional exponents?
5. How do we handle zero exponents in simplifications?

**Tip:** When applying exponent rules, it's crucial to keep track of both the base and the exponent to avoid errors, especially with negative or fractional exponents!

Simplify the expression (3n^5 / 2m^7)^3 and write the answer using only positive exponents.

Learn how to simplify the expression (3n^5 / 2m^7)^3 and express the result using only positive exponents. This problem involves exponent rules, including the power of a quotient and product. Suitable for students in Grades 9-11.

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