To simplify the expression \((-9m^{13}n^{13})^{-3}\), follow these steps:

### Step 1: Apply the negative exponent rule
The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). Therefore, the expression becomes:

\[
(-9m^{13}n^{13})^{-3} = \frac{1}{(-9m^{13}n^{13})^3}
\]

### Step 2: Raise each term inside the parentheses to the power of 3
Now, raise \(-9\), \(m^{13}\), and \(n^{13}\) to the power of 3 individually:

\[
(-9)^3 = -729, \quad (m^{13})^3 = m^{39}, \quad (n^{13})^3 = n^{39}
\]

Thus, the expression becomes:

\[
\frac{1}{-729m^{39}n^{39}}
\]

### Final Answer:
\[
\boxed{\frac{1}{-729m^{39}n^{39}}}
\]

Would you like more details on any step?

### Related Questions:
1. How do you handle exponents with negative bases?
2. What is the power of a product rule?
3. How do you simplify expressions with multiple variables?
4. Can a fraction's denominator be negative?
5. What happens when you raise a negative number to an even or odd power?

### Tip:
Always handle negative exponents by applying the reciprocal rule before simplifying powers.

Learn how to simplify the expression (-9m^13n^13)^-3 using the negative exponent rule. This solution involves raising each term to the power of 3 and applying exponent rules for variables. Suitable for students in Grades 9-12.

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