Is it true that: $e^{\theta^{N}} = e^{N\theta} \:\: \forall N \in \mathbb{N}$?

Question

I have a simple exponential power question about e (mathematical constant), is it true that: $$e^{\theta^{N}} = e^{N\theta} \:\: \forall N \in \mathbb{N}$$

Answer 1

$(e^\theta)^N=e^{N\theta}\neq e^{(\theta^N)}$

Answer 2

Unlike addition and multiplication, exponentiation is not an associative operation. That is, in general, we do not have $$(a^b)^c=a^{(b^c)},$$ so we can only let $$a^{b^c}$$ represent one of these. By convention, we tend to let $$a^{b^c}=a^{(b^c)},$$ so while $$(a^b)^c=a^{bc}$$ for natural numbers $c$, we don't in general have $$a^{b^c}=a^{bc}$$