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The given expression is:

\[
\frac{x^{2n+1} \cdot y^{n+1}}{z^{2n}} \cdot \frac{y^{1-n} z^3}{x^{n-1}}
\]

We can simplify it step by step:

### Step 1: Combine the two fractions
\[
\frac{x^{2n+1} \cdot y^{n+1}}{z^{2n}} \cdot \frac{y^{1-n} z^3}{x^{n-1}} = \frac{x^{2n+1} \cdot y^{n+1} \cdot y^{1-n} \cdot z^3}{z^{2n} \cdot x^{n-1}}
\]

### Step 2: Simplify the terms involving \(x\), \(y\), and \(z\):

- **For \(x\):**
  \[
  x^{2n+1} \cdot x^{-(n-1)} = x^{2n+1-(n-1)} = x^{2n+1-n+1} = x^{n+2}
  \]

- **For \(y\):**
  \[
  y^{n+1} \cdot y^{1-n} = y^{(n+1) + (1-n)} = y^{2}
  \]

- **For \(z\):**
  \[
  \frac{z^3}{z^{2n}} = z^{3-2n}
  \]

### Step 3: Final simplified expression
Now putting everything together:
\[
x^{n+2} \cdot y^2 \cdot z^{3-2n}
\]

Thus, the simplified expression is:
\[
x^{n+2} \cdot y^2 \cdot z^{3-2n}
\]

Would you like more details or explanations on any part of this process?

Here are some related questions:
1. How do you simplify expressions with different exponents?
2. Can you show an example with specific values for \(n\) to verify the result?
3. What are the rules for combining powers of variables?
4. How does negative exponents affect simplification?
5. How does this simplification change if we have additional terms?

**Tip**: Always combine like terms step-by-step to avoid errors, especially when exponents involve both addition and subtraction.

Simplify the expression: (x^(2n+1) * y^(n+1)) / z^(2n) * (y^(1-n) * z^3) / x^(n-1)

Learn how to simplify the expression (x^(2n+1) * y^(n+1)) / z^(2n) * (y^(1-n) * z^3) / x^(n-1) step-by-step. This problem covers key algebraic concepts like exponentiation rules and the simplification of expressions with multiple variables. Suitable for students in Grades 9-11.

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