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Use the properties of exponents to simplify the expression. Express your answer using positive exponents.

The image provided contains a mathematical problem involving the simplification of an expression using the properties of exponents, and the final expression must use only positive exponents. 

Let me transcribe and simplify the expression provided.

The expression seems to be:

\[
\frac{3^{4m} \cdot 9^{n-1} \cdot 81^{m+2}}{27^{2m-n} \cdot 243^{1-3m+n}}
\]

**Step 1: Express using prime bases**
1. \(9 = 3^2\)
2. \(27 = 3^3\)
3. \(81 = 3^4\)
4. \(243 = 3^5\)

Substitute these into the expression:

\[
\frac{3^{4m} \cdot (3^2)^{n-1} \cdot (3^4)^{m+2}}{(3^3)^{2m-n} \cdot (3^5)^{1-3m+n}}
\]

**Step 2: Simplify the expression**

- Simplify each term by multiplying the exponents:

\[
\frac{3^{4m} \cdot 3^{2(n-1)} \cdot 3^{4(m+2)}}{3^{3(2m-n)} \cdot 3^{5(1-3m+n)}}
\]

This becomes:

\[
\frac{3^{4m} \cdot 3^{2n-2} \cdot 3^{4m+8}}{3^{6m-3n} \cdot 3^{5-15m+5n}}
\]

**Step 3: Combine the exponents**

Combine all the exponents in the numerator and the denominator:

- Numerator: \( 3^{4m + 2n - 2 + 4m + 8} = 3^{8m + 2n + 6} \)
- Denominator: \( 3^{6m - 3n + 5 - 15m + 5n} = 3^{-9m + 2n + 5} \)

**Step 4: Subtract the exponents**

Since the bases are the same, subtract the exponent in the denominator from the exponent in the numerator:

\[
3^{(8m + 2n + 6) - (-9m + 2n + 5)} = 3^{8m + 2n + 6 + 9m - 2n - 5} = 3^{17m + 1}
\]

Thus, the simplified expression is:

\[
\boxed{3^{17m + 1}}
\]

Would you like further details on any specific step or have any other questions?

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Here are five related questions that can expand on this topic:

1. How can you simplify expressions that involve fractional exponents?
2. What are the common mistakes to avoid when simplifying expressions with exponents?
3. How do you simplify expressions that have a mix of positive and negative exponents?
4. Can you demonstrate how to simplify a similar expression with variables in the exponents?
5. How do you approach simplifying expressions with more than two levels of exponents?

**Tip:** Always start by rewriting each term with a common base before simplifying exponent expressions.

Learn how to simplify the expression involving exponents using exponent rules. This problem covers key concepts such as using properties of exponents and converting expressions to positive exponents. Suitable for high school students.

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