We are asked to simplify the expression $3b \cdot (2b^2)^3$.

1. Problem Description

We are asked to simplify the expression $3b \cdot (2b^2)^3$.

2. Solution Steps

First, we simplify the term $(2b^2)^3$ using the power of a product rule, which states that $(ab)^n = a^n b^n$.

$(2b^2)^3 = 2^3 (b^2)^3$

$2^3 = 2 \cdot 2 \cdot 2 = 8$.

Next, we use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

$(b^2)^3 = b^{2 \cdot 3} = b^6$.

So, $(2b^2)^3 = 8b^6$.

Now, substitute this back into the original expression:

$3b \cdot (2b^2)^3 = 3b \cdot 8b^6$.

Multiply the coefficients and add the exponents of $b$:

$3b \cdot 8b^6 = (3 \cdot 8) (b^1 \cdot b^6) = 24b^{1+6} = 24b^7$.

3. Final Answer

$24b^7$