Problem: Understanding negative exponents and 0^0

Problem

1. $b$ and $c$ are both positive integers and $b>c$ . Prove that $\frac{2^b}{2^c}=2^{b-c}$ (power rule 2)
2. We want power rule 2 to be valid even when $b=c$ . Explain why this forces us to define $2^0$ as $1$
3. We want power rule 2 to be valid even when $b<c$ . Explain why this forces us to define $2^{-1}$ as $1/2$ (and similarly $2^{-2}$ as $1/2^2$ and so on)
4. Calculate $0^{0.00001}$ and ${0.00001}^0$ . Explain why it makes more sense to leave $0^0$ undefined.