Add water and sugar to a large punch bowl, stirring to dissolve sugar completely. Add root beer extract and stir to combine. Taste and add any more sugar or extract needed.
Make sure you are mixing the root beer in a well-ventilated area. Carefully break up the dry ice pieces into very small pieces.
Add the dry ice pieces using gloves or tongs, stir into the water, sugar, and extract mixture making sure to keep the ice from sticking to the bowl and to melt the dry ice.
The mixture will start to bubble and the fog will pour out and over the bowl. Make sure ice is completely melted before ladling into cups or mugs.
To be extra safe, you can pour the mixture through a mesh strainer before stirring to filter out any small pieces of dry ice.