Yes, there is. There is a law of exponentiation. # The definition of exponentiation shows us it is a product, but the law of exponentiation shows us it is a sum. # Law of exponentiation is confirmed by a general formula to be expressed by words. # From there one sees exponentiation to be constructed by the two constants: base constant and exponent constant. It is the property of exponentiation. Author Đoàn Đức Nhuận (in vietnamese)

There isn't one. It is an algebraic operation, not an arithmetic one and is called exponentiation.

Squaring. Doubling is only multiplying a number by 2, whereas, squaring is multiplying a number by itself :)

That may be a mis-spelling of "exponentiation".

Squaring the ends of metal in a lathe is called farting