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Let's simplify the given expression step by step.

The expression is:

\[
\frac{\left(\frac{4x^5}{y^{4n}}\right)^2 \left(\frac{y^{-2}}{4x^2}\right)^3}{\left(\frac{5x^{2n}}{(xy)^4}\right)^2}
\]

### Step 1: Simplify each part individually

#### 1. \(\left(\frac{4x^5}{y^{4n}}\right)^2\)
\[
\left(\frac{4x^5}{y^{4n}}\right)^2 = \frac{(4^2)(x^{5 \cdot 2})}{(y^{4n})^2} = \frac{16x^{10}}{y^{8n}}
\]

#### 2. \(\left(\frac{y^{-2}}{4x^2}\right)^3\)
\[
\left(\frac{y^{-2}}{4x^2}\right)^3 = \frac{(y^{-2})^3}{(4x^2)^3} = \frac{y^{-6}}{64x^6}
\]

#### 3. \(\left(\frac{5x^{2n}}{(xy)^4}\right)^2\)
\[
\left(\frac{5x^{2n}}{(xy)^4}\right)^2 = \frac{(5^2)(x^{2n \cdot 2})}{(xy)^8} = \frac{25x^{4n}}{x^8y^8} = \frac{25x^{4n-8}}{y^8}
\]

### Step 2: Multiply the simplified numerators

\[
\frac{16x^{10}}{y^{8n}} \times \frac{y^{-6}}{64x^6} = \frac{16x^{10}y^{-6}}{64x^6y^{8n}} = \frac{x^{10-6}y^{-6-8n}}{64} = \frac{x^4y^{-6-8n}}{64}
\]

### Step 3: Combine the entire expression

Now, put this over the denominator:

\[
\frac{\frac{x^4y^{-6-8n}}{64}}{\frac{25x^{4n-8}}{y^8}} = \frac{x^4y^{-6-8n} \cdot y^8}{64 \cdot 25x^{4n-8}} = \frac{x^{4+8-4n}y^{-6-8n+8}}{1600} = \frac{x^{12-4n}y^{2-8n}}{1600}
\]

### Final Simplified Expression

The simplified expression is:

\[
\frac{x^{12-4n}y^{2-8n}}{1600}
\]

Would you like more details on this simplification process, or do you have any other questions?

### Related Questions:
1. What happens if \( n = 0 \) in the final expression?
2. How would the expression change if the exponents were not integers?
3. How do you handle negative exponents in a complex fraction?
4. How can the expression be interpreted for large values of \( n \)?
5. What is the significance of simplifying such algebraic expressions?

### Tip:
Always simplify expressions inside parentheses first before raising them to any power; it makes the calculation much easier.

Learn how to simplify the given algebraic expression involving fractional exponents step-by-step. This problem covers algebraic concepts such as exponentiation and fraction simplification, suitable for advanced high school students.

Simplify Fractional Exponent Expression: \(\frac{\left(\frac{4x^5}{y^{4n}}\right)^2 \left(\frac{y^{-2}}{4x^2}\right)^3}{\left(\frac{5x^{2n}}{(xy)^4}\right)^2}\)

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