To simplify the expression $(2ab^3)^2 (-3ac^2)^3$, we follow these steps:

1. Expand each term:

   • $(2ab^3)^2 = 2^2 (a^2)(b^3)^2 = 4a^2b^6$
   • $(-3ac^2)^3 = (-3)^3 (a^3)(c^2)^3 = -27a^3c^6$

2. Combine the results:

   • Now we multiply the two expanded results: $(4a^2b^6)(-27a^3c^6)$

3. Multiply the coefficients and combine the variables:

   • Coefficients: $4 \cdot -27 = -108$
   • Variables:
     • $a^2 \cdot a^3 = a^{2+3} = a^5$
     • $b^6$ remains as is.
     • $c^6$ remains as is.

4. Final expression: $-108a^5b^6c^6$

Thus, the simplified expression is $-108a^5b^6c^6$.

If you have any questions or need further details, feel free to ask!

Related Questions

1. How do you apply the power of a product property in this expression?
2. Can you provide examples of similar expressions for practice?
3. What are the general rules for multiplying polynomial expressions?
4. How would the expression change if there were additional variables?
5. What is the significance of negative coefficients in algebraic expressions?

Tip

Always keep track of the exponent rules when expanding and simplifying expressions.