https://file.aiquickdraw.com/mathbot/file/17277092713660qo6arrz.jpg

The expression to simplify from the image is:

\[
\frac{2 + 16m^{-1}}{1 - 64m^{-2}}
\]

Let's work through this step by step:

### Step 1: Simplify the denominator
The term \(1 - 64m^{-2}\) can be factored. Notice that \(64m^{-2}\) is a perfect square:

\[
1 - 64m^{-2} = (1 - 8m^{-1})(1 + 8m^{-1})
\]

### Step 2: Simplify the numerator
The numerator is \(2 + 16m^{-1}\), which cannot be factored further.

### Step 3: Express the complex fraction
Now we have the fraction in this form:

\[
\frac{2 + 16m^{-1}}{(1 - 8m^{-1})(1 + 8m^{-1})}
\]

This is the simplified version of the given complex fraction. If you'd like further simplifications, let me know.

Would you like a detailed explanation of any step, or have questions on any part?

Here are 5 relative questions that expand this topic:
1. How do you factor expressions involving negative exponents?
2. What is the general strategy for simplifying complex fractions?
3. Can complex fractions always be reduced to simpler forms?
4. How do operations with negative exponents differ from positive ones?
5. How can you identify special factorizations like difference of squares?

**Tip:** When working with complex fractions, always check if the numerator or denominator can be factored to make simplifications easier.

Simplify the complex fraction: (2 + 16m^(-1)) / (1 - 64m^(-2))

This solution demonstrates how to simplify the complex fraction (2 + 16m^(-1)) / (1 - 64m^(-2)) by factoring and using negative exponents. Suitable for students in Grades 9-12, the problem focuses on simplifying rational expressions and applying the difference of squares theorem.

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