This isn't a situation where brute force is the optimal solution. You can't really blow up an asteroid like in the movies -- and if you could, it could actually make things worse, because all that mass would still be on the same collision course for Earth, and it would come down spread out far more widely. The only real solution is to divert the asteroid so it misses the planet. Which isn't that hard to do. It only takes the Earth about 7 minutes to traverse its own diameter as it moves through its orbit, so you just need to change the asteroid's course or speed a tiny bit so it crosses our orbit a few minutes later or earlier. This is the benefit of detection systems, because the earlier we find these objects, the easier it is to deflect them; not only do you have more time to alter an asteroid's course, but the farther away it is, the less of a course change you need to impart to get it to miss.
I have to add to my comments above, since I wasn't completely right. When dealing with objects of a few dozen to a few hundred meters across -- potential city-killers like the Chelyabinsk object rather than planet-killers -- it could be viable to vaporize them completely given enough advance warning. There's a proposed laser system that would do just that, discussed in this Centauri Dreams post
(which happens to make the same Rendezvous with Rama
reference I made just above).
Although I have to question whether it would be necessary to use the lasers to vaporize them completely, when it would be more efficient just to nudge them off course. Maybe the idea is to prevent them from ever being threats again?