I don't see how it does. The problem is not power, but momentum. You can beam power, but where does your reaction mass come from? Producing the required levels of power is relatively practical, as these things go, but reaction mass is subject to the nasty exponential growth of the rocket equation.
If you're talking something like solar sail off of a laser, the numbers I'm finding[0] say 300MW per N of thrust. I don't think you'll be travelling to the stars anytime soon on that.
I think you're talking about relativistic mass, which is irrelevant here. When it comes to rocket propulsion, reaction mass is everything. See the Rocket Equation[0]
By the law of Momentum, you cannot change the velocity of an object without pushing against, and changing the velocity, of another object. Vehicles on Earth mostly push against the planet, and the water and air on it. In space, there is nothing, so you must bring your own mass to push against if you want to change your velocity. But this sets up a rather inconvenient exponential situation, because the mass you expel to accelerate now must accelerate not only your vehicle, but also all of the rest of the mass that you're going to expel later to accelerate further.
It's why chemical rockets traveling to Earth orbit have about 20x more propellant mass than the mass that's going to actually reach orbit. The situation gets exponentially worse the higher you want to take the final craft velocity, which is why all interplanetary missions have gotten around mainly on gravitational slingshots. Increasing the rocket exhaust velocity helps, but only in a linear fashion. When it comes to reaching reasonable interstellar velocities, all the numbers I've seen say that it's wildly impractical even with fusion or anti-matter level exhaust velocities. Like mass ratios in the neighborhood of 4,000,000, as I calculated above, which is still nowhere near fast enough.
No, a particle accelerator accelerates particles close to the speed of light. The large hadron collider gets protons to 7TeV which = 99.9999991%c They also gain mass from this but if your spaceship only needs to hit .1c then that's more or less irrelevant.
Ion thrusters basically are low energy particle accelerators already. The problem, aside from energy consumption, is scaling up the thrust. You can't just cram twice as many particles through your particle accelerator without scaling up the mass of the thruster itself, or decreasing the exit velocity. And it does you very little good if your rocket has the fuel to accelerate to .1 c, if it takes you 10,000 years to do so.
If you're talking something like solar sail off of a laser, the numbers I'm finding[0] say 300MW per N of thrust. I don't think you'll be travelling to the stars anytime soon on that.
[0] http://www.projectrho.com/public_html/rocket/reactionlessdri...