Damage resistance: If you have resistance to a certain type of damage, you take half damage. No need to roll a save, you can't drop it any further unless it's a save for no damage (i.e. evasion) you can't drop from half to 1/4th.
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Not true. Damage resistance halves the damage you take. That's all. If a fireball hits for 20 damage and you fail the save, you take 20 damage, whivh is halved to 10.
If you succeed on the save, you take 10 damage which is halved to 5.