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Barbarian Rage grants resistance to bludgeoning, piercing, and slashing damage. On the other hand, Uncanny Dodge allows you to halve any single attack using your reaction.

In the case of a multiclassed character, do they stack?

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They do stack

There is nothing limiting you from taking 1/4th the damage of any attack given the right circumstances. For instance, if you have resistance to fire damage and you have a dexterity saving throw success against a fireball, you take half damage from the it and another half of that due to resistance, netting you 1/4th damage.

So yes, you can halve the incoming damage, which would then be halved again by resistance.

You could, in theory, continue to halve the damage forever as long as you can create the circumstances to allow that math to take place, since there is no rule for minimum damage, percentage or otherwise.

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  • \$\begingroup\$ +1 for mentioning the existing resistance + save for half as precedence. \$\endgroup\$ – Christopher May 9 '16 at 4:06
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Yes, they stack

The only limitation that I can find with regards to damage on PH 197 states that resistance doesn't stack.

Since only one of the abilities you're asking about confer resistance —the other just modifies damage—there is no stacking conflict.

Note that uncanny dodge is only usable one per round, since it requires your reaction, so cannot stack on more than one attack.

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