To use up, form an origianl meaning of drawing out (“ex”) humours.

• 08.108 Poor old Bombur was so exhausted –

“exhaust, v.” OED Online. Oxford University Press, June 2015. Web. 27 July 2015.

To call (“claim”) out.

• 06.002 Good heavens!’ he exclaimed.
• 12.077 exclaimed Bilbo aloud,
• 18.014 he exclaimed.

“exclaim, v.” OED Online. Oxford University Press, June 2015. Web. 27 July 2015.

In this word, the prefix “ex-” is an intensifier, not a prepositional prefix.  So, “to intensely aggregate a pile of things.”

• 01.094 poetical exaggeration applied to any hobbit,

“exaggerate, v.” OED Online. Oxford University Press, June 2015. Web. 27 July 2015.