How to clean a Toshiba laptop fan without taking it apart

The idea would be to dislodge the dirt/dust actually. Blow the compressor into the exhaust vent, then blow it into the fan intake, then repeat a few times. Also try to blow the air into any other vented intakes. Repeat it a few times and the temps should drop significantly if it's as dirty as you say. By repeating the steps you should be able to get the dislodged dirt/dust outside the laptop without needing to open it.