To simplify the expression

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

we'll go through the following steps:

Step 1: Simplify the Base of the Fraction Inside the Parentheses

The base expression inside the parentheses is:

$\frac{2X^{-4} \cdot Y^3}{x^{-5} y^5 \cdot X^2 y^3}$

Separate Each Component

1. For $X$:

   • In the numerator: $X^{-4}$
   • In the denominator: $X^2$
   • Combining the exponents: $X^{-4 - 2} = X^{-6}$

2. For $Y$:

   • In the numerator: $Y^3$
   • In the denominator: $y^5 \cdot y^3 = y^8$
   • Combining the exponents for $Y$: $Y^{3 - 8} = Y^{-5}$

3. For the constant 2:

   • The constant 2 remains in the numerator.

So, the expression inside the parentheses simplifies to:

$\frac{2 \cdot X^{-6} \cdot Y^{-5}}{1} = 2X^{-6}Y^{-5}$

Step 2: Apply the Exponent of 5

Now we need to raise the entire simplified expression to the power of 5:

$(2X^{-6}Y^{-5})^5$

Using the power rule $(a \cdot b)^n = a^n \cdot b^n$, we get:

1. $2^5 = 32$
2. $X^{-6 \cdot 5} = X^{-30}$
3. $Y^{-5 \cdot 5} = Y^{-25}$

Thus, the expression becomes:

$32X^{-30}Y^{-25}$

Step 3: Rewrite with Positive Exponents

To express this with positive exponents, we move $X^{-30}$ and $Y^{-25}$ to the denominator:

$\frac{32}{X^{30}Y^{25}}$

Final Answer

The simplified expression is:

$\frac{32}{X^{30}Y^{25}}$

Would you like further details or have any questions?

---

Here are 5 related questions to deepen your understanding:

1. How do you simplify expressions with negative exponents in the numerator and denominator?
2. What is the general rule for simplifying expressions with variables raised to powers in fractions?
3. How would the process change if the exponent was negative instead of positive?
4. How can we verify the solution by substituting specific values for $X$ and $Y$?
5. How would the solution differ if we had additional terms with different variables?

Tip: Remember that applying the power rule to a fraction requires raising both the numerator and denominator terms to that power individually.