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Say you have a +1 composite (+2) longbow. It would cost 2600, 300 for masterwork, 300 as a composite +2, 2000 for +1 enhancement.

I am wanting to upgrade the bow to have the weapon property adaptive, which is a 1000 cost enhancement. Which would bring the cost upto 3600? Or would it replace the composite +2 and only cost an additional 800?

Do I end up with an adaptive composite +2 bow? How would the two abilities work together? I must have a minimum +2 str to use the bow, but it gives me a damage bonus equal to str mod?

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This is a case of specific trumps general. In general, the PFRPG core book says the following about a composite longbow:

All composite bows are made with a particular strength rating (that is, each requires a minimum Strength modifier to use with proficiency).

However, Ultimate Equipment's adaptive property (an adaptive composite longbow being the more specific item) says:

An adaptive bow responds to the strength of its wielder, acting as a bow with a strength rating equal to its wielder's Strength bonus.

The adaptive property trumps the innate property of the bow. It explicitly says it changes the strength rating to match the strength of the user and allows you to use less than your Strength if you desire.

If you're making the item from scratch, you may be wasting money by paying for a strength rating higher than +0. However, if you take your adaptive bow into an area that suppresses magic, the adaptive property ceases working and it reverts to whatever it physically is.

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