system
Given the following post, respond as if you're commenting on Reddit:
human
how would you explain x^-1 and x^1/y?

How would you explain taking a number to the negative power?
I don't understand how that would be the reciprocal of x!!
I also don't get taking a number to the reciprocal of something like 4^1/2 how is that the square root of 4?
gpt
Well, for the first one think about it this way. If you multiply two numbers with the same base raised to an exponent, you simply add the exponents: a^x * a^y = a^x+y. Now what happens if you divide by a? Lets say a^3 / a = a^2. But that's exactly the same thing as saying a^3 * a^-1 = a^2. So by comparison a^-1 must be the reciprocal of a , if we follow the rules of multiplying two numbers with the same base.

It's the same kind of idea with raising something to the fraction of a power. See if you can prove it to yourself.