Let's analyze and simplify the given expression. The expression is:

$\frac{m^{-1} n^{-2}}{m^8 n^{-4} \cdot m^5 n}$

Step-by-Step Simplification:

1. Combine the terms in the denominator:

   • $m^8 \cdot m^5 = m^{8+5} = m^{13}$
   • $n^{-4} \cdot n = n^{-4+1} = n^{-3}$

   So, the denominator becomes $m^{13} n^{-3}$.

2. Rewrite the entire expression:

$\frac{m^{-1} n^{-2}}{m^{13} n^{-3}}$

3. Simplify the exponents by using the property of exponents:

   For $m$: $\frac{m^{-1}}{m^{13}} = m^{-1 - 13} = m^{-14}$

   For $n$: $\frac{n^{-2}}{n^{-3}} = n^{-2 - (-3)} = n^{-2 + 3} = n^1 = n$

4. Final simplified expression: $\frac{m^{-1} n^{-2}}{m^{13} n^{-3}} = m^{-14} n$

   Since the problem asks to express the answer using positive exponents, convert $m^{-14}$ into a positive exponent:

$\boxed{\frac{n}{m^{14}}}$

This is the simplified answer using positive exponents.

Would you like any more details or explanations? Here are some related questions to expand on the concept:

1. What are the basic exponent rules for simplifying expressions?
2. How can you multiply and divide powers of the same base?
3. How do you convert negative exponents to positive exponents?
4. What is the general approach to simplifying algebraic fractions?
5. How do you handle expressions with multiple variables and exponents?

Tip: When dealing with negative exponents, remember that $x^{-a} = \frac{1}{x^a}$, which helps you turn negative exponents into positive ones.