[math-fun] Least set of reals containing Q+ and closed under both multiplication and exponentiation

This may well be trivial, since I haven't thought about it more than a few seconds. But let Q^ denote the least set of reals containing Q+ and closed under both multiplication and exponentiation. I.e., Q^ is the union of all Q_n, n >= 0, where * Q_0 := Q+ * Q_(n+1) = (Q_n)^(Q_n) * (Q_n)^(Q_n), n >= 1. where for any subsets X and Y of R+, X op Y := {x op y | x in X and Y in Y} for op in {*,^}. Question: Is Q^ closed under addition? --Dan