The problem asks us to simplify the expression $8a^2 : \frac{4a^2}{b}$. This is equivalent to dividing $8a^2$ by the fraction $\frac{4a^2}{b}$.

1. Problem Description

The problem asks us to simplify the expression $8a^2 : \frac{4a^2}{b}$. This is equivalent to dividing $8a^2$ by the fraction $\frac{4a^2}{b}$.

2. Solution Steps

To divide by a fraction, we multiply by its reciprocal. The reciprocal of $\frac{4a^2}{b}$ is $\frac{b}{4a^2}$. So we have:

$8a^2 : \frac{4a^2}{b} = 8a^2 \times \frac{b}{4a^2}$

Now, we can simplify by canceling common factors.

$8a^2 \times \frac{b}{4a^2} = \frac{8a^2b}{4a^2}$

We can divide both the numerator and the denominator by $4a^2$.

$\frac{8a^2b}{4a^2} = \frac{8}{4} \times \frac{a^2}{a^2} \times b$

$\frac{8}{4} = 2$ and $\frac{a^2}{a^2} = 1$, so

$\frac{8a^2b}{4a^2} = 2 \times 1 \times b = 2b$

3. Final Answer

$2b$