The word radical has a lot of interesting definitions, but radical arithmetic doesn't actually refer to arithmetic that favors drastic political, economic, or social reforms. We're talking about doing arithmetic with expressions that contain radical symbols. Sorry, you can put those signs down.
We've seen two special types of expressions so far: polynomial expressions, and rational expressions. Now we'll add one more special type, because things are funnier in threes: a radical expression is any expression with one or more radical signs in it. Another way to put this is that a radical expression has at least one radical term, or a term with at least one radical in it. Yet one more way to put this, because things are funnier in threes, is that a radical expression radiates with radicalocity.
Hmm, not so much. So much for the "rule of threes."