The problem asks to simplify the following expression:

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

Let's simplify this step by step.

Step 1: Simplify inside the parentheses

We have:

$\frac{x^3 y^6}{x^5 y^{-2}}$

Using the properties of exponents, subtract the exponents for both $x$ and $y$:

• For $x$, we subtract the exponents: $3 - 5 = -2$
• For $y$, we subtract the exponents: $6 - (-2) = 6 + 2 = 8$

Thus, we now have:

$x^{-2} y^8$

Step 2: Apply the power of 4

Now, apply the exponent of 4 to the simplified expression:

$\left( x^{-2} y^8 \right)^4$

This results in multiplying each exponent by 4:

• $x^{-2} \to x^{-8}$
• $y^8 \to y^{32}$

Thus, the expression becomes:

$x^{-8} y^{32}$

Final Answer:

$\boxed{x^{-8} y^{32}}$

This is the simplified form of the given expression.

Do you have any questions or need further details?

Here are some related questions for deeper understanding:

1. How do you apply the laws of exponents when dividing powers with the same base?
2. What happens when you raise a power to another power?
3. How would the answer change if the base was negative?
4. Can you express $x^{-8}$ without using negative exponents?
5. How would you simplify a similar expression involving cube roots instead of powers?

Tip: Always simplify inside parentheses first before applying any outside exponents or other operations.