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The problem asks which expression correctly decomposes the fraction \( \frac{36}{8} \) into a product of powers.

First, let's simplify the fraction \( \frac{36}{8} \):
\[
\frac{36}{8} = \frac{36 \div 4}{8 \div 4} = \frac{9}{2}
\]
So, \( \frac{36}{8} = \frac{9}{2} \).

Now, we decompose \( 9 \) and \( 2 \) into powers:
- \( 9 = 3^2 \)
- \( 2 = 2^1 \)

Thus, \( \frac{9}{2} \) becomes:
\[
\frac{9}{2} = \frac{3^2}{2^1} = 3^2 \times 2^{-1}
\]

Additionally, since no other factors are involved, the correct decomposition would include powers of 3 and 2.

### Evaluating the answer choices:
1. \( 2^2 \times 4^{-1} \times 2^{-2} \): This isn't correct because it introduces powers of 4, which aren't relevant.
2. \( 3^2 \times 2^{-1} \times 1^{-2} \): This matches our decomposition, but the factor \( 1^{-2} \) is unnecessary, as \( 1^n = 1 \) for any power. Thus, this is incorrect due to the extraneous \( 1^{-2} \).
3. \( 6^2 \times 2^{-1} \times 2^{-2} \): This includes \( 6^2 \), which is not part of the decomposition of \( 9 \) and \( 2 \), making it incorrect.

### Conclusion:
None of the given choices perfectly match the correct decomposition of \( \frac{36}{8} \). However, choice 2 is closest to the correct answer but includes an unnecessary factor.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How do we simplify a fraction like \( \frac{36}{8} \)?
2. Why do we decompose numbers into powers in math problems like this?
3. What rules govern the exponents when working with fractions?
4. How does multiplying and dividing powers of the same base work?
5. What would the decomposition look like for other similar fractions?

**Tip**: Always ensure to simplify the numbers in a fraction before breaking them down into powers.

Which shows 36/8 decomposed into a product of powers?

Understand how to decompose the fraction 36/8 into a product of powers using exponentiation and fraction simplification. This problem illustrates how powers can be applied in the context of fractions, suitable for Grades 6-8.

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