Well obtaining slackware CDs would be an excellent 1st step
One way to do it is to format your mandriva partition and install slackware on it:
Once you are in the slack installer you will have a step where you choose on which partitions to install slackware, choose the current linux partition (as usual good practice is to backup the data prior) format it and viola fresh space for slackware!
If you haven't split /home and /usr/local mount points from / then now is a good time to do so, then if you want to reinstall/upgrade again later you won't need to restore all your user config and programs.
Just a note, there may be a much easier method for doing this, I don't know.