You'd be multiplying the square roots, not exponentiating. Yeah, I don't know how practical a construction would be overall for someone who seriously went at i

You'd be multiplying the square roots, not exponentiating. Yeah, I don't know how practical a construction would be overall for someone who seriously went at it; I was mainly moved to answer because this objection that constructions can't express transcendental numbers just doesn't seem relevant -- digitally you don't keep infinite precision either. Square roots and multiplying are two of the simplest geometric constructions (geometric mean and taking a proportion), compass and straightedge operations can be accurate, you can construct it enlarged and then scale it down, and finally the precision you need to aim for at the end is bounded. A potentially fun project idea. Think of it as a kind of retrocomputing: how might Euclid or Archimedes have designed a slide rule?