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.