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!