A binary operation $*$ is defined on the set of real numbers $R$ by $a * b = 4a^2b$. We need to evaluate $\frac{1}{4} * \frac{1}{2}$.

1. Problem Description

A binary operation $*$ is defined on the set of real numbers $R$ by $a * b = 4a^2b$. We need to evaluate $\frac{1}{4} * \frac{1}{2}$.

2. Solution Steps

Given the definition of the binary operation $*$, we have:

$a * b = 4a^2b$

We need to evaluate $\frac{1}{4} * \frac{1}{2}$. Substituting $a = \frac{1}{4}$ and $b = \frac{1}{2}$ into the definition of the operation, we get:

$\frac{1}{4} * \frac{1}{2} = 4 \left(\frac{1}{4}\right)^2 \left(\frac{1}{2}\right)$

$\frac{1}{4} * \frac{1}{2} = 4 \left(\frac{1}{16}\right) \left(\frac{1}{2}\right)$

$\frac{1}{4} * \frac{1}{2} = 4 \left(\frac{1}{32}\right)$

$\frac{1}{4} * \frac{1}{2} = \frac{4}{32}$

$\frac{1}{4} * \frac{1}{2} = \frac{1}{8}$

3. Final Answer

$\frac{1}{8}$