We need to simplify the given expression:

$\frac{\left( x^{-5} y^3 \right)^4 \cdot \left( x \cdot y^6 \right)^2}{\left( x^5 y^4 \right)^4}.$

Step 1: Expand each term using the power rule $(a^m)^n = a^{m \cdot n}$.

1. Numerator:

   • Expand $(x^{-5} y^3)^4$:
     $(x^{-5} y^3)^4 = x^{-5 \cdot 4} y^{3 \cdot 4} = x^{-20} y^{12}.$
   • Expand $(x \cdot y^6)^2$:
     $(x \cdot y^6)^2 = x^{1 \cdot 2} y^{6 \cdot 2} = x^2 y^{12}.$
     Combining these two, the numerator becomes:
     $x^{-20} y^{12} \cdot x^2 y^{12}.$
     Using the product rule $x^a \cdot x^b = x^{a+b}$:
     $x^{-20 + 2} y^{12 + 12} = x^{-18} y^{24}.$

2. Denominator:
   Expand $(x^5 y^4)^4$:
   $(x^5 y^4)^4 = x^{5 \cdot 4} y^{4 \cdot 4} = x^{20} y^{16}.$

Step 2: Combine numerator and denominator.

The simplified expression becomes:
$\frac{x^{-18} y^{24}}{x^{20} y^{16}}.$

Using the quotient rule $\frac{x^a}{x^b} = x^{a-b}$:

1. For $x$-terms:
   $x^{-18 - 20} = x^{-38}.$
2. For $y$-terms:
   $y^{24 - 16} = y^8.$

Thus, the simplified result is:
$x^{-38} y^8.$

Final Answer:

$\frac{\left( x^{-5} y^3 \right)^4 \cdot \left( x \cdot y^6 \right)^2}{\left( x^5 y^4 \right)^4} = x^{-38} y^8.$

Would you like me to explain any part of this further or solve a similar problem?

Related Questions:

1. How do you simplify an expression involving negative exponents?
2. What is the product rule for exponents, and how is it applied?
3. How does the power of a power rule work for exponents?
4. How do you handle variables in both the numerator and denominator when simplifying?
5. What happens when you have terms with zero exponents?

Tip:

Always simplify step by step, following exponent rules: product, quotient, and power rules!