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Let's solve the given expression step by step. The expression is:

\[
\frac{2^{-2}}{1 + \frac{3^{-1}}{1 + 3^{-1}}}
\]

### Step 1: Simplify the exponentials
- \( 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \)
- \( 3^{-1} = \frac{1}{3} \)

### Step 2: Substitute these values back into the expression
The expression becomes:

\[
\frac{\frac{1}{4}}{1 + \frac{\frac{1}{3}}{1 + \frac{1}{3}}}
\]

### Step 3: Simplify the denominator of the large fraction
First, simplify the inner fraction:

\[
1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}
\]

Now substitute this back:

\[
\frac{\frac{1}{4}}{1 + \frac{1}{3} \times \frac{3}{4}}
\]

### Step 4: Simplify further
The next step is to multiply the fractions:

\[
\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}
\]

So the expression now becomes:

\[
\frac{\frac{1}{4}}{1 + \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{4}{4} + \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{5}{4}}
\]

### Step 5: Simplify the entire expression
Finally, simplify the fraction by multiplying by the reciprocal:

\[
\frac{\frac{1}{4}}{\frac{5}{4}} = \frac{1}{4} \times \frac{4}{5} = \frac{1}{5}
\]

### Conclusion
The value of the expression is \(\frac{1}{5}\).

The correct answer is:

**\(\frac{1}{5}\)**

Would you like more details or have any questions? 

Here are five questions related to this content:

1. What is the value of \(5^{-2}\)?
2. How do you convert a negative exponent to a positive one?
3. What are the properties of fractions involving exponents?
4. How do you simplify complex fractions?
5. What is the significance of reciprocal in fraction operations?

**Tip:** When working with fractions involving exponents, always simplify the exponent first before moving on to other operations.

Learn how to solve complex exponential and fractional operations step-by-step. This problem involves simplifying expressions with negative exponents and complex fractions, suitable for students in Grades 6-8.

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