Well, if you want to re-tile or grout your bathroom it sounds like a good mix. But one cup of arrowroot is a lot!

It should take you a long time to use that quantity of this mixture. Just a little chunk here or there in a sauce or a soup will serve you well.

Sadassa has it right! make into smaller batches for continued use. Think Beurre manie. It should work fine for pan sauces or gravies. Good advice Sadassa Ulna

Make a thickened sauce. Both arrow root and butter are considered thickening agents in sauces. Although I don't use arrow root for a thickening agent It is used for that purpose. I do however mount pan sauces when finishing with butter. I imagine you should be a able to use the combination however unusual as it may seem to thicken a sauce or a soup. I'm not sure if one batch of anything can use up your amount though.An entire cup of arrow root is a large quantity. Also, I'm not sure if it will work but if you just want to save the butter you might try clarifying to see if you can get the arrow root to float for skimming. Not sure if the butter would thicken from the arrow root and never reach a liquid state. That's just some thoughts.

Some gluten-free baked goods use corn starch in lieu of wheat flour. I am imagine arrowroot is similar to corn starch? Or, I have never tried what I am about to suggest, but it might be an alternative to throwing it all out: make little blobs like you would drop cookies, freeze them, and keep them on hand for thickening gravy