Explain how exponents <1 are roots. It appears sensible, since it's the opposite operation. But I can't find any actual logical basis for it. The definition of exponentiation is multiplying something by itself n amount of times. This is fine for positive integers, but what does it actually mean to multiply something a half time? Or any fractional times? How do you perform this operation in a way which is consistent with the definition?

The obvious answer seems to be that since the exponent is how many times you have the base in a series of multiplications, then a half exponent means you have it a half time. That's not a root, and the resulting curve a^n would be a series of lines connecting the integer powers.

I'm not looking for extrinsic proofs or any way of showing that it must be so therefore it is, miss me with that shit. I'm looking for a definition for multiplying something by itself a half time.