I don't know whether or not is rational or not, but there is an interesting proof in which this quantity figures.
Problem: Can be rational if a and b are irrational?
Consider . If x is rational then we have our example. If not, and x is irrational, then consider , and we again have an example of rational with irrational a, b. So whether x is rational or not, we have the required example.
This is an example of (to me) an absolutely infuriating proof. We know either x or is an example of rational with irrational a and b, but we don't know which, and the proof gives us no clue!
This proof is discussed in "The Princeton Companion to Mathematics".