Principal root of 3+4i

Question

Is there a neat way of writing the principal root of 3+4i? I have an answer, but it is very ugly. Thanks for any help in advance.

Further, I think it would be helpful if you mentioned what your current answer is and why you think it's ugly — davidlowryduda, May 16 '14 at 04:34

score 0 · Answer 1 · edited Nov 23 '17 at 19:47

0

$3+ 4i= 5e^{ \arctan(4/3)}$. The principal $p$th root is $\sqrt[p]{5}e^{\frac{\arctan(4/3)}{p}i}$

edited Nov 23 '17 at 19:47

Davide Giraudo

answered Nov 23 '17 at 19:27

user247327

score 0 · Accepted Answer · answered May 16 '14 at 04:33

0

$(2+i)^2=4-1+4i=3+4i$

You can show it by $(a+ib)^2=3+4i\implies a^2-b^2=3,2ab=4$

Polar form : $\sqrt 5 e^{i\arctan(1/2)}$

answered May 16 '14 at 04:33

evil999man

1

Ok, but only assuming the question is about the square root. AFAICT the OP didn't specify :-) – Jyrki Lahtonen May 16 '14 at 04:42

2 Answers2