We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequencies