We are given the expression:

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

Let’s break it down step by step:

Step 1: Simplify the fraction inside the parentheses

$\frac{x^{-3} y^{-4}}{x y^2}$ To simplify this, we apply the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$. This gives:

$\frac{x^{-3}}{x} = x^{-3-1} = x^{-4}$ $\frac{y^{-4}}{y^2} = y^{-4-2} = y^{-6}$

Now, the expression becomes: $x^{-4} y^{-6}$

Step 2: Apply the exponent of -2

We now raise $x^{-4} y^{-6}$ to the power of $-2$: $(x^{-4} y^{-6})^{-2} = x^{-4 \times -2} y^{-6 \times -2} = x^8 y^{12}$

Step 3: Apply the exponent of 3

Now, raise the result to the power of 3: $(x^8 y^{12})^3 = x^{8 \times 3} y^{12 \times 3} = x^{24} y^{36}$

Final Answer:

$x^{24} y^{36}$

This is the simplified expression.

Would you like more details or explanations about any step?

Here are 5 related questions for further exploration:

1. How do you simplify expressions involving negative exponents?
2. What are the rules for raising a power to another power in exponents?
3. How can the laws of exponents be applied in solving algebraic equations?
4. What is the significance of using parentheses in exponentiation?
5. How do you simplify complex rational expressions with multiple variables?

Tip: Always simplify the inner expressions first before applying outer exponents in compound expressions.