How to get the Mammoth Tusk Powder

The briar hearts in this quest are apparently prone to glitches. I found this thread where people are discussing the various ways in which they were able to break this quest. In some cases, when they got notification that they needed to get that specific briar heart, all the briar hearts they already had became quest items.

I think the best you can do is load an earlier save, or hang on to this quest until Bethesda patches it to work properly. Even so, you might not be able to finish it unless they create some way to reset your progress in it or otherwise fix the quest if it is already in progress.

There is currently no way to break down mammoth tusks to get powdered tusks. The only way to obtain some is to purchase it in an alchemy store or to find it as loot. Source

Or if you are desperate for some and are on the PC, you can do it via the console:

player.additem 6bc10 [amount you want]