I have this question about exponentiation. How can one prove that $(a^b)^c$ = $a^{(bc)}$ for rational and negative exponents?
Firstly, we need a basis of what a rational and negative exponent are.
So, for these purposes, I define rational exponentiation, for a number to the power of the fraction $\frac mn$, so $a^{\frac mn}$ as being $(\sqrt[n]a)^m$ and a negative exponent $a^{(-x)}$ as being $\frac 1a^x$.
How can one prove that these hold? Also, how can one prove that this holds for any number of exponents (e.g. $((a^b)^c)^d$ = $a^{(bcd)}$?
Also, how can one prove this holds for "mixed" exponents, like $(a^3)^\frac 23$, or $(a^3)^{-2}$?
Thank you. I'm aware I posted this prior, but the post was of very low quality, I didn't define either and I felt as if it was best to rewrite it, and make it clear.