In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$?

Say we start with $\Bbb N$: $\Bbb N$ is closed under addition and multiplication. But then we decide we'd like a number system that's closed under negation as well, so we construct $\Bbb Z$. Great. But then we decide we'd like to extend this number system further to be closed under division and so we construct $\Bbb Q$. The next step is closure under exponentiation - but when we construct that number system, we don't get $\Bbb R$, we get a subset of $\Bbb C$ which I'll call $\Bbb Q_{\exp}$.

Now clearly when constructing $\Bbb R$ from $\Bbb Q$ we do gain completeness, but our gain is then analytic, not necessarily algebraic. Do we gain any algebraic advantage in constructing $\Bbb R$ from $\Bbb Q$ similar to what we get at each of the other constructions I mentioned?