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.