Let's simplify the given expression step by step.

The expression is:

$\left(\frac{6a^3b^{-2}}{27a^{-2}b^{-4}}\right)^{-3}$

Step 1: Simplify the expression inside the parentheses.

First, divide the coefficients and apply the laws of exponents.

$\frac{6a^3b^{-2}}{27a^{-2}b^{-4}} = \frac{6}{27} \cdot \frac{a^3}{a^{-2}} \cdot \frac{b^{-2}}{b^{-4}}$

Now, simplify each part:

1. Coefficients: $\frac{6}{27} = \frac{2}{9}$

2. For $a$: $\frac{a^3}{a^{-2}} = a^{3 - (-2)} = a^{3 + 2} = a^5$

3. For $b$: $\frac{b^{-2}}{b^{-4}} = b^{-2 - (-4)} = b^{-2 + 4} = b^2$

So, the expression inside the parentheses simplifies to:

$\frac{2}{9} \cdot a^5 \cdot b^2 = \frac{2a^5b^2}{9}$

Step 2: Apply the exponent $-3$ to the simplified expression.

$\left(\frac{2a^5b^2}{9}\right)^{-3} = \frac{9^3}{(2a^5b^2)^3}$

Now calculate each part:

1. For the coefficient: $\frac{9^3}{2^3} = \frac{729}{8}$

2. For $a^5$: $(a^5)^3 = a^{15}$

3. For $b^2$: $(b^2)^3 = b^6$

So, the expression becomes:

$\frac{729}{8} \cdot \frac{1}{a^{15}b^6} = \frac{729}{8a^{15}b^6}$

Final Answer:

$\boxed{\frac{729}{8a^{15}b^6}}$

Would you like more details or have any questions?

Here are 5 related questions you could explore:

1. What are the properties of exponents?
2. How does the order of operations affect simplifying expressions?
3. How do you handle negative exponents in expressions?
4. What is the process for simplifying complex fractions?
5. How do you apply the distributive property with exponents?

Tip: Always simplify the expression inside parentheses before applying any external exponents!