https://r2.erweima.ai/mathbot/image/1725420039436ider2f07.jpg

To simplify the expression given in the image:

\[
\left(\frac{3mn^4}{mn^3}\right)^{-2}
\]

and express it with only positive exponents, follow these steps:

### Step 1: Simplify the expression inside the parentheses

Simplify the fraction by canceling out common terms:

1. **Variable \(m\)**:
   \[
   \frac{m}{m} = 1
   \]
   So, the \(m\) terms cancel out.

2. **Variable \(n\)**:
   \[
   \frac{n^4}{n^3} = n^{4-3} = n^1 = n
   \]

So, the simplified expression inside the parentheses becomes:

\[
3n
\]

### Step 2: Apply the exponent \(-2\)

Now, apply the exponent \(-2\) to the expression:

\[
\left(3n\right)^{-2} = \frac{1}{(3n)^2}
\]

This can be expanded further:

\[
\frac{1}{9n^2}
\]

### Final Answer:

\[
\frac{1}{9n^2}
\]

Would you like any further details or explanations on this? Here are some related questions:

1. How would you simplify an expression if there were more variables involved?
2. What are the steps to simplify expressions with fractional exponents?
3. How do you handle expressions with multiple terms inside the parentheses?
4. How does the process change when the exponents inside the expression are negative?
5. What is the impact of raising a fraction to a negative exponent?

**Tip:** When simplifying expressions, always cancel out like terms first before applying any exponents. This makes the process easier and less prone to mistakes.

Learn how to simplify expressions with negative exponents by converting them to positive exponents. This problem involves the fundamental concept of exponents and their manipulation. Suitable for high school students.

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