To find the product of \( 82 \) and \( \frac{5}{10} \), we first simplify \( \frac{5}{10} \):
\[ \frac{5}{10} = \frac{1}{2} \]
Now, we compute the product:
\[ 82 \times \frac{1}{2} = \frac{82}{2} = 41 \]
Next, we need to express \( 41 \) in simplest radical form. The number \( 41 \) is a whole number and can also be expressed as a square root:
\[ 41 = \sqrt{41^2} = \sqrt{1681} \]
Since \( 41 \) is not a perfect square (it is a prime number), the simplest radical form of \( 41 \) is just \( 41 \) itself. Thus, the final answer is:
\[ \boxed{41} \]