Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers?

Question

Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows:

$$ w = a+bi \\ z = c+di $$

Can we analogically write

$$ \sqrt[w]{z} = z^\frac{1}{w} \qquad \rightarrow \qquad \sqrt[a+bi]{c+di} = (c+di)^\frac{1}{a+bi} $$

from what we know about real numbers?

I couldn't find this in the internet and currently I don't have comprehensive a textbook about complex numbers under my hand. This may be a very simple question for most of you, but I don't know the answer. — hkBattousai, Apr 27 '13 at 11:47
Related: http://math.stackexchange.com/questions/201991/for-what-values-alpha-for-complex-z-lnz-alpha-alpha-lnz/202164#202164 — lab bhattacharjee, Apr 27 '13 at 12:03

score 2 · Accepted Answer · answered Apr 27 '13 at 12:53

2

$\sqrt[w]{z}$ is answer of $z=x^w $ so

$x=\sqrt[w]{z}$ and $x^{w*\frac{1}{w}}=z^{\frac{1}{w}}$ so

$\sqrt[w]{z}=z^{\frac{1}{w}}$

answered Apr 27 '13 at 12:53

Somaye

1 Answers1