To rewrite \(8 \frac{1}{2}\) using a root, we can first convert it into an improper fraction:
\[ 8 \frac{1}{2} = \frac{17}{2} \]
Now, we look to represent this value using square roots. Since \(8 = 4 \times 2\), we can express \(8\) as \( \sqrt{64} \) because \(64 = 8^2\).
Thus, we can rewrite \(8 \frac{1}{2}\) as follows:
\[ 8 \frac{1}{2} = 8 + \frac{1}{2} = \sqrt{64} + \frac{1}{2} \]
Alternatively, if we want to express the entire value \(8 \frac{1}{2}\) using a single root, we can say:
\[ 8 \frac{1}{2} = \frac{17}{2} = \sqrt{\left(\frac{17}{2}\right)^2} = \sqrt{\frac{289}{4}} = \frac{\sqrt{289}}{2} \]
Thus \( \frac{\sqrt{289}}{2}\) also represents \(8 \frac{1}{2}\).
Since there are several ways to represent this, other forms might include using \(2\) as the square root denoting the fractional part separately, but the key idea is that \(8 \frac{1}{2}\) can be expressed in various ways involving roots depending on how you want to formulate it.