I'm a crude word for a BM, often associated with a single large piece, and sorry for any unwanted visuals. I'm also a three letter mathematical term, sometimes annoying, defining an exponent in multiplication. I'm a piece of tree, so please solve the word; an easy three letter term.

What irony, or trick of eye, To ascribe to things, what they deny, Such that a thing becomes defined, By what it simply won't abide, And so becomes identified, By what it chose to cast aside, Not by what it holds inside, But that without, which we have spied

Hidden numbers, only counted with each delicious bite, Fuel for life's ceaseless motions, in the day or night. A unit of measurement, though difficult to keep in sight Each gives you energy, regardless if taken in misery or delight

I exist only when you stop looking at me directly. I am born from agreement, but I collapse under proof. I can control crowds without force, erase truth without lies and survive centuries without a body. I am not a word but I live inside words.

Each clue below points to a different four letter word. All four words use the same letters. All answers accepted: Skin torch Keep open? Sit it down. Recent tour?