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: $$e^{\theta^{N}} = e^{N\theta} \:\: \forall N \in \mathbb{N}$$

$(e^θ)^N = e^{Nθ}$, but $e^{3^3} = e^{27} ≠ e^9 = e^{3·3}$ by the injectivity of the exponential function. To what extent do you need an explanation for the first statement? — k.stm, May 05 '13 at 13:53
For any number $\ne 0,\pm1,$ this is not true. In general $(a^b)^c=a^{bc}$. Your question demands $\theta^N=N\theta$ — lab bhattacharjee, May 05 '13 at 13:53
@labbhattacharjee Oh okay, you cleared that up for me. Thank you! — Dexter, May 05 '13 at 13:59

score 1 · Accepted Answer · edited May 05 '13 at 13:56

1

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

edited May 05 '13 at 13:56

mercio

50,180

answered May 05 '13 at 13:54

Ma Ming

7,482

Thanks, I was unsure, but now I see that you need brackets raised to N. – Dexter May 05 '13 at 13:56

score 0 · Answer 2 · answered May 05 '13 at 14:03

0

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 $$a^{b^c}=a^{bc}$$

answered May 05 '13 at 14:03

Cameron Buie

102,994

Thanks for that, just doing some work and was just unsure about it. – Dexter May 05 '13 at 14:06

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

2 Answers2