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

1. Problem Description

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

2. Solution Steps

We are given the binary operation $a * b = a + a^2b$.

We want to evaluate $\frac{1}{4} * \frac{1}{2}$.

In this case, $a = \frac{1}{4}$ and $b = \frac{1}{2}$.

Substituting these values into the expression, we get

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

$(\frac{1}{4})^2 = \frac{1}{16}$

So, $\frac{1}{4} * \frac{1}{2} = \frac{1}{4} + \frac{1}{16} \cdot \frac{1}{2}$

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

To add these fractions, we need a common denominator, which is

3

2. $\frac{1}{4} = \frac{8}{32}$

So, $\frac{1}{4} * \frac{1}{2} = \frac{8}{32} + \frac{1}{32}$

$\frac{1}{4} * \frac{1}{2} = \frac{9}{32}$

3. Final Answer

$\frac{9}{32}$