To simplify the given expression $(ab^2c)^{-3} \times b^{-3} \times 3$, let's break it down step by step:

Step 1: Simplify $(ab^2c)^{-3}$

The negative exponent indicates that we need to take the reciprocal of the base and then apply the exponent.

$(ab^2c)^{-3} = \frac{1}{(ab^2c)^3}$

Now, apply the exponent of 3 to each factor inside the parentheses:

$\frac{1}{(ab^2c)^3} = \frac{1}{a^3(b^2)^3c^3} = \frac{1}{a^3b^6c^3}$

Step 2: Multiply by $b^{-3}$ and 3

Next, multiply the result by $b^{-3}$ and 3:

$\frac{1}{a^3b^6c^3} \times b^{-3} \times 3$

Recall that multiplying by $b^{-3}$ is the same as multiplying by $\frac{1}{b^3}$:

$= \frac{3 \times 1 \times 1}{a^3b^6c^3 \times b^3} = \frac{3}{a^3b^{6+3}c^3} = \frac{3}{a^3b^9c^3}$

Final Answer

The simplified expression is:

$\frac{3}{a^3b^9c^3}$

Let me know if you need any more details or have further questions. Here are some related questions to consider:

1. What is the process to simplify expressions with negative exponents?
2. How do you simplify expressions with multiple variables and exponents?
3. How can you apply exponent rules to more complex expressions?
4. What is the difference between negative and positive exponents in simplification?
5. How do you handle fractional exponents in expressions?

Tip: When dealing with exponents, always remember to apply the exponent rules step by step to avoid mistakes.