https://file.aiquickdraw.com/mathbot/file/1725915237856aq8e1azq.jpg

The expression shown in the image is:

\[
\left( \frac{m^2}{n^9} \right)^3
\]

We are asked to simplify this expression assuming all variables are non-zero.

### Step-by-Step Simplification:
1. Apply the power of a quotient rule: \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\). This gives us:
   \[
   \frac{(m^2)^3}{(n^9)^3}
   \]

2. Apply the power of a power rule: \((a^m)^n = a^{m \cdot n}\). This gives:
   \[
   \frac{m^{2 \cdot 3}}{n^{9 \cdot 3}} = \frac{m^6}{n^{27}}
   \]

So, the simplified expression is:

\[
\frac{m^6}{n^{27}}
\]

Now, let's match this result with the given multiple choice answers. The correct option is:

\[
\boxed{\frac{m^6}{n^{27}}}
\]

---

Would you like more details or explanations on any part of this solution? Here are five related questions that expand on this:

1. What are some common rules for simplifying expressions with exponents?
2. How do you simplify an expression involving a power of a quotient?
3. What is the difference between the power of a power rule and the power of a product rule?
4. Can exponent rules apply when variables are negative or zero?
5. How do you simplify more complex expressions that involve multiple exponents and variables?

**Tip**: Always check the order of operations when working with exponents to avoid mistakes. Apply the exponent rules carefully, especially when powers are nested.

Simplify: Assume all variables are non-zero. \(\left( \frac{m^2}{n^9} \right)^3\)

Learn how to simplify the expression \(\left( \frac{m^2}{n^9} \right)^3\) using exponent rules like the power of a quotient and power of a power. This step-by-step guide is suitable for students in Grades 8-10.

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