The problem in the image asks for the simplified form of the expression:

$\frac{(2m^5n^{-5})}{(32m^7n^{-1})}$

Step-by-step solution:

1. Simplify the coefficients: $\frac{2}{32} = \frac{1}{16}$ So the expression becomes: $\frac{m^5n^{-5}}{m^7n^{-1}} \cdot \frac{1}{16}$

2. Simplify the powers of $m$: Using the quotient rule of exponents $\frac{a^m}{a^n} = a^{m-n}$: $\frac{m^5}{m^7} = m^{5-7} = m^{-2}$

3. Simplify the powers of $n$: Similarly, for $n$: $\frac{n^{-5}}{n^{-1}} = n^{-5 - (-1)} = n^{-5 + 1} = n^{-4}$

4. Combine the results: So far, we have: $\frac{1}{16} \cdot m^{-2} \cdot n^{-4} = \frac{m^{-2}n^{-4}}{16}$

   This can be written as: $\frac{1}{16m^2n^4}$

The correct answer is $\boxed{\frac{1}{16m^2n^4}}$, but the choices suggest we need to match a given format. Since this doesn't directly appear, the most likely simplification based on the image choices would be $\boxed{\frac{1}{16m^2n^4}}$.

Would you like further clarification, or to ask any additional questions?

Related Questions:

1. How do you apply the quotient rule of exponents?
2. What is the general form of simplifying expressions involving negative exponents?
3. How can fractions be combined with variables in algebra?
4. Why do we subtract exponents when dividing like bases?
5. How does this example apply to more complex expressions?

Tip:

Always apply exponent rules carefully, particularly when dealing with negative exponents or fractions.