We are given the expression:

$\frac{(3x^2y - 2)^3}{(9xy^3)^3}$

To simplify this, we will break down the steps:

Step 1: Expand the denominator

The denominator is $(9xy^3)^3$. We can distribute the exponent 3 across the terms inside the parentheses:

$(9xy^3)^3 = 9^3 \cdot (xy^3)^3 = 729 \cdot x^3 \cdot (y^3)^3$ Since $(y^3)^3 = y^9$, the denominator becomes:

$729x^3y^9$

Step 2: Rewrite the expression

Now the expression becomes:

$\frac{(3x^2y - 2)^3}{729x^3y^9}$

Step 3: Simplification

Unfortunately, the numerator $(3x^2y - 2)^3$ cannot be factored further in a simple form since it's a binomial cubed. You would need to expand it using the binomial theorem if you wish to express it in terms of powers of $x$ and $y$.

The fully simplified version, unless expanded further, is:

$\frac{(3x^2y - 2)^3}{729x^3y^9}$

Would you like to expand the numerator using the binomial theorem? Let me know if you'd like further details on this.

Further Questions

1. Would you like to simplify the numerator completely by expanding it?
2. Do you need help with any specific step or operation?
3. How familiar are you with binomial expansions?
4. Would you like to explore a related problem involving exponents?
5. Do you prefer working with more complex algebraic fractions?

Tip

When simplifying expressions with exponents, always apply the power rule carefully to each term in both the numerator and denominator before combining or reducing terms.