A binary operation $*$ is defined on the set of real numbers $R$ by $m * n = m + n - \frac{1}{2}n$. We are asked to evaluate $(1+\sqrt{3}) * (1-\sqrt{3})$.

1. Problem Description

A binary operation $*$ is defined on the set of real numbers $R$ by $m * n = m + n - \frac{1}{2}n$. We are asked to evaluate $(1+\sqrt{3}) * (1-\sqrt{3})$.

2. Solution Steps

Let $m = 1+\sqrt{3}$ and $n = 1-\sqrt{3}$.

Using the definition of the binary operation $*$, we have

$m * n = m + n - \frac{1}{2}n$

Substitute $m = 1+\sqrt{3}$ and $n = 1-\sqrt{3}$ into the equation:

$(1+\sqrt{3}) * (1-\sqrt{3}) = (1+\sqrt{3}) + (1-\sqrt{3}) - \frac{1}{2}(1-\sqrt{3})$

$= 1 + \sqrt{3} + 1 - \sqrt{3} - \frac{1}{2} + \frac{\sqrt{3}}{2}$

$= 1+1 - \frac{1}{2} + \sqrt{3} - \sqrt{3} + \frac{\sqrt{3}}{2}$

$= 2 - \frac{1}{2} + \frac{\sqrt{3}}{2}$

$= \frac{4}{2} - \frac{1}{2} + \frac{\sqrt{3}}{2}$

$= \frac{3}{2} + \frac{\sqrt{3}}{2}$

$= \frac{3+\sqrt{3}}{2}$

3. Final Answer

$\frac{3+\sqrt{3}}{2}$