Here's a question that's going to come up soon in my campaign. We're playing reign of winter where about 90% of the enemies are vulnerable to fire and last session when we leveled our spellcaster picked up Empower Spell.
So my question is what happens when an empowered fire spell hits a target weak to fire how is the damage calculated, is it just doubled (adding the two 50% boosts together) or do you multiply the total damage after empower by 50%?
The order of operations is you figure out the damage of the spell, then apply it to the target and then adjust for resistance or vulnerability. Both of these steps are discrete, so you would round each time.
So if I roll 13 damage on my spell, I will add 6 to that if it is empowered, for a total of 19 damage. If the target is vulnerable, it would take 50% more damage (+9) for a total of 27. If I rolled a 14 instead, it would be 14 + 7 = 21 + 10 for a total of 31.
Also apparently we've been using the empower spell effect wrong all this time, we've been rolling the damage a second time and then halving the second roll instead of just multiplying the total
Mathematically the average would be the same. The shape of the curve would slightly change (if you roll really well or really poorly on the damage roll, the extra is also effected, whereas 2 rolls are more likely to average out) but the difference is going to be minimal.
While it isn't the way the rules say to do it, your method doesn't particularly change the game and if you enjoy that more it would be fine.
Everything posted above is wrong.
Multiplying Damage: Sometimes you multiply damage by some factor, such as on a critical hit. Roll the damage (with all modifiers) multiple times and total the results.
Note: When you multiply damage more than once, each multiplier works off the original, unmultiplied damage. So if you are asked to double the damage twice, the end result is three times the normal damage.
Exception: Extra damage dice over and above a weapon's normal damage are never multiplied.
Multipliers are additive, so %150+%150=%200
Neither Empower Spell nor vulnerabilities are worded as multipliers. Each says you increase by 50%, and are calculated after the damage has been rolled (unlike the rule you quoted which multiplies how many dice you roll, and is therefore applied before the roll by necessity).
Vulnerability means the creature takes half again as much (+50%) damage as normal from that energy type, regardless of whether a saving throw is allowed or if the save is a success or failure.
"half again as much damage as normal" is what is specified, the word "multiply" never appears. Now, we just need to figure out what is "normal" damage for an empowered spell. I think you would be hard-pressed to argue that "normal" damage for an empowered spell is the unincreased damage, since it is normal that empower spell increases damage.
All variable, numeric effects of an empowered spell are increased by half, including bonuses to those dice rolls.
Again, no mention of the word "multiply." We are told to increase the damage (and all other variable numeric effects the spell may have) by half, not roll extra dice.
Order of operations would be apply multiplications before rolling (per your quote), then roll, then empower spell, then vulnerability.
Arguing that these use the multiplication rules you quoted is nonsensical in the case of odd amounts of damage dice -- you can't roll a die 1.5 times (it'd round down to 1), which would make things that multiply by 1.5x mostly worthless.
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Unless you have a method for calculating empower that doesn't involve multiplication, it uses the rules for multiplication, (it being multiplication and all).
Unless you have a suorce you failed to mention
Empower don't say that it multiply the result by 150%.
Benefit: All variable, numeric effects of an empowered spell are increased by half, including bonuses to those dice rolls.
You add half of the damage you will generate rolling the dices, as explained in maximize description:
An empowered, maximized spell gains the separate benefits of each feat: the maximum result plus half the normally rolled result.
It it was a multiplication a empowered, maximized spell would deal 150% of its maximum damage.
Quantum Steve wrote:
Unless you have a method for calculating empower that doesn't involve multiplication, it uses the rules for multiplication
X+(X/2) requires no multiplication.
I'd just like to point out that there is a transitive relationship between x+(x/2) and x*1.5. The rule about multiplication clearly stands here.
For example, if an effect causes someone to take half damage from an attack, is that multiplication? Dividing by two is the same as multiplying by half, after all. (If passing a reflex save and empowerment are both multiplication, then passing a reflex save against an empowered spell would cause you to take 100% damage, not 75%.)
The intent of the multiplication-by-addition rule was to stop combined effects (like a critical-hit lance-charge) from increasing damage exponentially. Similarly, the purpose of the maximised empowered spell rule is to prevent the two effects together from increasing damage to an insane degree. I suspect, therefore, that the intent is that vulnerable creatures take 200% damage from empowered spells, not 225%, because that's slightly more balanced. But who knows?
The question is if it is linear or multiplicative.
Is the end damage 1+0.5+0.5 = 2 or 1.5*1.5 = 2.25
Since empowered spell alters the number of dice, let's say from 10d6 to 15d6, that total damage would then be increased by 50% from the vulnerability. This can be approximated by rolling 10d6 and multiplying by 2.25.