To rewrite the fraction \( \frac{8}{12} \) using a root, first simplify the fraction:
\[ \frac{8}{12} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{2}{3} \]
To express \( \frac{2}{3} \) using a root, we can use square roots:
\[ \frac{2}{3} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{\sqrt{4}}{\sqrt{3^2}} = \frac{\sqrt{4}}{\sqrt{3 \cdot 3}} = \sqrt{\frac{4}{9}} = \sqrt{\frac{8}{12}} \]
So, \( \frac{8}{12} \) can be rewritten as \( \sqrt{\frac{8}{12}} = \frac{\sqrt{8}}{\sqrt{12}} \), but none of the provided options directly represent this expression.
Among the given options, \( 8 - \sqrt{8} \) and \( \sqrt{8} \) are not valid forms of \( \frac{8}{12} \) since they yield numerically different results.
To recap: \( \frac{8}{12} \) can be understood in terms of roots, but the closest valid representation among given options is not entirely represented in that list. If you choose to rewrite \( \frac{8}{12} \) as \( \sqrt{\frac{8}{12}} \)—although not explicitly listed in the options—it does represent the concept.