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.