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My barbarian is going to receive the Great Weapon Master feat. Looking over it, I have a question about the -5 to hit mentioned in the feat.

The second benefit of the Great Weapon Master feat (PHB, pg. 167):

Before you make a melee attack with a heavy weapon that you are proficient with, you can choose to take a -5 penalty to the attack roll. If the attack hits, you add +10 to the attack's damage.

Currently, I have a greataxe with a +5 modifier to the attack roll (+3 Str mod and +2 proficiency bonus). My question regards the -5 to hit from the feat. Do I subtract it from the +5 of the attack roll with the axe, giving me +0 attack bonus overall? Or does my overall modifier on the attack roll become a -5?

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You would combine them with any other pertinent modifiers and, assuming none else, end up at a final attack modifier of +0.

The penalty from Great Weapon Master doesn't replace, overwrite, or modify any other modifiers you might have. Few if any modifiers do, and they would almost certainly state outright that they did so, as that would be highly unusual.

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Generally, modifiers do not replace each other

-5 penalty to the attack roll

"Penalty" is a game term, meaning negative modifier. The positive modifier is called "bonus":

A class feature, a spell, a particular circumstance, or some other effect might give a bonus or penalty to the check

Modifiers do not replace each other, but are meant to be added:

To make an attack roll, roll a d20 and add the appropriate modifiers.

This is relevant for any features, not only the Great Weapon Master feat. You add all your modifiers to the d20 roll, in your case, that would be d20 + 3 (STR) + 2 (proficiency) - 5 (GWM).

Keep in mind, you never add your proficiency bonus twice, unless a feature specifically says so.

See this section of Basic Rules for more info.

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