Let's simplify the expression:

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

Step 1: Apply the negative exponent

A negative exponent means to take the reciprocal of the expression and change the sign of the exponent. So, applying $(-3)$ to everything inside the parentheses:

$\left( \frac{a^3 \cdot b^{-2}}{c^6} \right)^{-3} = \frac{1}{\left( \frac{a^3 \cdot b^{-2}}{c^6} \right)^3}$

Step 2: Distribute the cube power across the terms

Now, raise each term inside the parentheses to the power of 3:

$\frac{1}{\left( a^3 \right)^3 \cdot \left( b^{-2} \right)^3 / \left( c^6 \right)^3}$

This simplifies to:

$\frac{1}{a^{9} \cdot b^{-6} / c^{18}}$

Step 3: Simplify the fraction

To simplify further, we can multiply the numerator and denominator by $c^{18}$ to eliminate the fraction:

$\frac{c^{18}}{a^{9} \cdot b^{-6}}$

Step 4: Apply the negative exponent to $b^{-6}$

Recall that $b^{-6} = \frac{1}{b^6}$, so the expression becomes:

$\frac{c^{18}}{a^{9} \cdot \frac{1}{b^6}} = \frac{c^{18} \cdot b^6}{a^9}$

Final Answer:

$\frac{c^{18} \cdot b^6}{a^9}$

Would you like further details or clarification on any of these steps? Here are 5 related questions to explore:

1. What happens if we change the power of $a$, $b$, or $c$ inside the parentheses?
2. How do negative exponents work in different types of algebraic expressions?
3. What is the rule for simplifying a fraction raised to a power?
4. Can this expression be simplified further using different mathematical properties?
5. How do fractional exponents affect an expression like this one?

Tip: When dealing with negative exponents, always remember that they flip the base to the denominator (or vice versa).