September 7th, 2019, 12:24 PM  #1 
Newbie Joined: Sep 2019 From: Avalon Posts: 1 Thanks: 0  i^i=1^(1/4)?
Now I do know that i^i=e^(pi/2+(2*n*pi)) and has infinite solutions where n is an integer plus the solutions are all real. But I recently asked this question to my friend and here is her reply: i^i=(i^(4i))^(1/4) =(i^((4)i))^(1/4) =(1^i)^(1/4) =1^(1/4) Now 1^(1/4) has four solutions i, i, 1 and 1 but none of them satisfy the solution of i^i. I can't find a mistake in her solution. Can somebody please help me out? This problem is bugging me for a while. Forgive me if I did some silly mistake. I am not really bright in maths. Last edited by skipjack; October 2nd, 2019 at 03:36 AM. 
September 7th, 2019, 01:32 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,636 Thanks: 1472 
The problem is that $1^i \neq 1$ $1^i = (e^{i2\pi})^i = e^{2\pi}$ $\left(1^i\right)^{1/4} = e^{\pi/2}$ 
September 7th, 2019, 06:41 PM  #3 
Senior Member Joined: Oct 2009 Posts: 905 Thanks: 354 
The law $a^{bc} = (a^b)^c$ is a dangerous one. It holds for real numbers, but not necessarily for complex numbers! Well, perhaps it holds in some form with multivalued functions, but it needs a proof.

September 28th, 2019, 09:34 PM  #4  
Senior Member Joined: Aug 2018 From: România Posts: 110 Thanks: 7  Quote:
I do not understand!From the "WolframAlpha" read: 1) https://www.wolframalpha.com/input/?i=1%5Ei%3D1 2) https://www.wolframalpha.com/input/?...e%5E%282pi%29 All the best, Integrator Last edited by Integrator; September 28th, 2019 at 09:36 PM.  
September 28th, 2019, 09:38 PM  #5  
Senior Member Joined: Aug 2018 From: România Posts: 110 Thanks: 7  Quote:
From the "WolframAlpha" read: https://www.wolframalpha.com/input/?i=i%5Ei.  1) How much do I do $\displaystyle 0^i$ where $\displaystyle i^2=1$ 2) How much do I do $\displaystyle 0^{i^i} $ where $\displaystyle i^2=1$ All the best, Integrator Last edited by skipjack; October 2nd, 2019 at 03:35 AM.  
October 2nd, 2019, 01:48 AM  #6 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 166 Thanks: 64 Math Focus: Area of Circle 
$$i^i=e^{ \pi /2}$$
